IvIODULI OF ALGEBRAIC SURFACES

F. Catanese - Unlverslta di Piss'

Contents of the Paper

Lecture I: Almost complex structures and the I<ursmishi family (§I-3)

Lecture lh Deformations of complex structures and Kuranishi's theorem

(§4-6)

Lecture III: Variations on the theme of deformations ( §7- i0)

Lecture IV: The classics/ case (§11-13)

Lecture V: Surfaces and their invariants (§14-15)

Lecture Vh Outline of the Enriques-Kodaira classification (§ 16-17)

Lecture VII: Surfaces of general type and their moduli (§ 18-20)

Lecture Vlll: Bihyperelliptic surfaces and properties of the moduli spaces

(§21-23)

Introduction

This paper reproduces with few changes the lectures I actually delivered at the

C. LM. E, Session in Montecatini, with the exception of most part of one lecture

where I talked at length about the geography of surfaces of general type: the reason

for riot including this material is that it is rather broadly covered in some survey

papers which will be published shortly ([Pe], [Ca 3], [Ca 2]).

Concerning my originM (too ambitious) intentions, conceived ~,hen I accepted

Eduardo Sernesi's kind invitation to lecture about moduli of surfaces, one may

notice some changes from the preliminary program: the topics "Existence of

moduli spaces for algebraic varieties" and "Moduli via periods" were not treated.

The first because of its broadness and complexity (I rea~lized it might require a

course on its own, while I mainly wanted to arrive to talk about surfaces of general

type), the second too because of its vastity and also for fear of overlapping with the

course by Donagi (which eventually did not treat period maps and variation of

# Amember of G.N.S.A.G.A. of C.N.R., and in the M.P.I. Research Project in Algebraic Geometry.

**The final version of the paper was completed during a visit of the author to the University of California, San Diego.

Hodge structures). Anyhow the first topic is exhaustively treated in Popp's lecture

notes ([Po]) and in the appendices to the second edition of Murnford's book on

Geometric Invariant Theory ([Mu Z] ), whereas the nicest applications of the theory

of variation of Hodge structures to moduli of surfaces are amply covered in the

book by B a r t h - P e t e r s - V a n de Ven ( [ :B-P-V]) .

Also, I mainly treated moduli of surfaces of general type, and fortunately

Seller lectured on the results of his thesis ([sei 1,2,3]) about the moduli of (polarized)

elliptic surfaces: Ihope his lecture notes are appearing in this volume.

Instead, the part on I(odaira-Spencer's theory of deformations and its connec-

tions with the classical theory of continuous systems started to gain a dominant

role after Igave a series of lectures at the Institute for Scientific Interchange

(I°S.I.) in Torino on this subject. In fact, after Zappa (cf. [Zp], [Iviu 3]) discov-

ered the first example of obstructed deformations, a smooth curve in an algebraic

surface, it was hard to justify most of the classical statements about moduli (and

in fact, cf. lecture four, some classical problems about completeness of the char-

acteristic system have a negative answer).

Interest in moduliwas revived only through the pioneering work of llodaira-

Spencer and later through Murnford's theory of geometric invariants. Murnford's

theory is mere algebraic and deals mostly with the problem of determining whether

a moduli space exists as an algebraic or projective variety, whereas the trans-

cendental theory of Kodaira and Spencer (in fact applied in an algebraic context by

Grothendiec~ and Artin) applies to the more general category of complex mani-

folds (or spaces), at the cost of producing only alocal theory. In both issues, it is

clear that it is not possible to have a good theory of moduli without imposing some

restriction on complex manifolds or algebraic varieties.

~rfaces of general type are a case when things work out well, and one would

like first rio investigate properties and structure of this moduli spaces, then to

draw from these results useful geometric consequences. It is my impression that

for these purposes (e.g. to count number of moduli) the Kodaira-Spencer theory is

by far more useful, and not difficult to apply in many concrete cases. In fact, it

seems that in most applications only elementary deformation theory is needed, and

that's one reason why these lecture notes cover very little of the more sophisti-

cated theory (Cfo §I0 for more details). The other reason is that the author is not

an expert in modern deformation theory and realized rather late about the existence

or importance of some literature on the subject: in particular we would like to

recommend the beautiful survey paper ( [Pa] ) by Palamodov on deformation of

complex spaces, whose historical introduction contains rather complete informa-

tion regarding the material treated in the first three lectures.

Since the style of the paper is already rather informal, we don't attempt any

discussion of the main ideas here in the introduction, and, before describing with

more detail the contents, we remark that the paper (according to the C. I°M° I~.

goals) is directed to and ought to be accessible to non specialists and to beginning

graduate students° Of course, reasons of space have obliged us to assume some

familiarity with the language of algebraic geometry, especially sheaves and linear

systerns~

Finally, in many points references are omitted for reasons of economy and

the lack of a quotation of some author's name (or paper) should not be interpreted

as any claim of originality on my side, or as an underestimation of some scientific

work.

§I-5 summarizes the essentials of the Kodaira-Spencer-Kuranishi results

needed in later sections, following existing treatments of the topic ([K-M], [Ku 3]),

whereas §6 is devoted to a single but enlightening example. §7 deals with defor-

mations of automorphisms, whereas §8-9 are devoted to Horikawa's theory of

deforrnations of holornorphic maps, with more emphasis to applications, such as

deformation of surfaces in 3-space, or of complete intersections, and include some

examples of everywhere obstructed deformations, due to Mumford and Kodaira.

§i0 is a "mea culpa '~ of the author for the topics he did not treat, §11-13 try to

compare Horikawa's and Schlessinger-Wahl's theory of embedded deformations,

whereas §IZ consists of a rewriting, with some simplifications of notation, of

Kodaira's paper ([Ko 3]) treating embedded deformations of surfaces with ordinary

singularities. §14-17 give a basic resurn~ on classification of surfaces and §18-19

are devoted to basic properties of surfaces of general type and a sketchy discus-

sion of Gieseker's theorem on their moduli spaces. §Z0-Z3 include a rough outline

of recent work of the author and a result of I. Reider: §Z0 deals with the number

of moduli of surfaces of general type, §ZZ outlines the deformation theory of

(~/Z) Z covers, §21 and 23 exhibit examples of moduli spaces with arbitrarily many

connected components having different dimensions, and discuss also the problem

whether the topological or the differentiable structure should be fixed.

Acknowled~ments : It is a pleasure to thank the Centro Internazionale Matematico Estivo and the Institute for Scientific Interchange of Torino for their invitations to lecture on the topics of these notes, and for their hospitality and support. I'm also very grateful to the University of California at San Diego for hospitality and support,

and especially to Ms. Annetta V/hiteman for her excellent typing.

LECTURE ONE: ALMOST COMPLEX STRUCTURES and the K U R A N I S H I F A M I L Y

I n t h i s l e c t u r e I w i l l r e v i e w t h e c o n s t r u c t i o n , d u e to K u r a n i s h i , o f t h e c o m p l e x

s t r u c t u r e s , o n a c o m p a c t c o m p l e x m a n i f o l d M , s u f f i c i e n t l y c l o s e to t h e g i v e n o n e .

T o do t h i s , o n e h a s t o u s e t h e n o t i o n o f a l m o s t c o m p l e x s t r u c t u r e s , o f i n t e g r a b l e

o n e s : i n a s e n s e o n e o f t h e m a i n t h e o r e m s , d u e to N e w I a n d e r a n d N i r e n b e r g , i s a

d i r e c t e x t e n s i o n o f a b a s i c t h e o r e m of d i f f e r e n t i a l g e o m e t r y , t h e t h e o r e m of

F r o b e n i u s .

§1 . A l m o s t c o m p l e x s t r u c t u r e s

L e t M b e a d i f f e r e n t i a b l e ( o r C ~ , i . e . r e a l a n a l y t i c ) m a n i f o l d o f d i m e n s i o n

e q u a l t o Zn, T M i t s r e a l t a n g e n t b u n d l e .

D e f i n i t i o n 1. 1.

T M ® C = T I'0

A n a l m o s t c o m p l e x s t r u c t u r e o n M i s t h e d a t u m of a s p l i t t i n g

TO, 1 w i t h T 1 , 0 T O , 1

Naturally, the splitting of T M ® f induces a splitting for the cornplexified cotan-

V ~ (~ = (TI,0) v (9 (T O' i) V ((TI'0) V is the annihilator of T O, l),and for gent bundle T M th

all the other tensors. In particular for the r exterior power of the cotangent

bundle, one has the decomposition Ar'T V ( M ~ C) = G AP(T I, 0)V~ Aq(T 0, I)V. p+q=r

We shall denote by gP'q the sheaf of C °~ sections of AP(T I' 0)V9 Aq(T 0' I) v

(resp. by ~P'q the sheaf of C ~ sections), by gr the sheaf of C = sections of r V

A (T M ® ¢ ) . ~:." %

The De Rham algebra is the differential graded algebra (g , d), where g = 2n @ gr , and d is the operator of exterior differentiation° For a function f, df E

r=0

gl, 0 • g0, 1 and one can wr{te accordingly df = Of + ~f; the problem is whether for

all forms q0 one can write d = @ + ~, with 8~ C p'q -~ gp+1,q, ~: £p,q -~ gp, q+l

(then one has @Z = ~2 = OR + ~8 = 0, since d 2= 0). Hence one poses the following

Definition I. 2. The given almost complex structure is integrable if

d(g p'q) c C p+l'q • gp, q+l

As a matter of fact, it is enough to verify this condition only for p = l, q = 0.

Lemma 1.3. The almost complex structure is integrable ~ d(gl,0) c g2,0 (9

g I, I [Hence another equivalent condition is: gl, 0 generates a differential ideal. ]

P r o o f • The q u e s t i o n b e i n g l o c a l , we c a n t a k e a l o c a l f r a m e fo r El , 0, i . e . s e c t i o n s

t~l . . . . . ~ n of g 1 , 0 w h o s e v a l u e s a r e l i n e a r l y i n d e p e n d e n t a t e a c h p o i n t ( l o c a l l y ,

C0 gl, 0 is a free module of rank n over , and ~i ..... ~ n ] is a basis). Our

weaker condition is thus that

(I,4) dLU = I ~ 3j A 'jJ + ~%/~ __~J) A ,'~-- B<Y ~By B V 8 aBy B v

(where ~0c~y~ and ~c~B~ are functions) since every a E gl, 0 can be written as

E n o~=l fc tu0~, and [tu~A tUy Ii ~ 8< y ~ n] is a local frame for g2,0 [~A ~y I

I ~ B,y ~ n] is a local frame for gl, l Now g0,1 = gl,0 hence

d(g0, I) c gl, 1 @ g0,2 and one verifies d(g p'q) c gp+l,q ~9 gp, q+l by induction n

A '2 + on p,q , since locally any ~ E g p'q canbe writtenas ~C~=l z1~ c~ Z n ~=l ~ A ~ , with ~ E gp-l,q ~9 E gp, q-i Q.E.D.

0t Ct ' q£

At this stage, one has to observe that if M is a complex manifold, then

(TV)I,0 = (T 1,0)v is generated (by definition !) by the differentials df of holo-

morphic functions Cat least locally, if one has a chart (z l ..... Zn): U -~ C n ,

dz I ..... dz give a frame for (TV) I'0). Conversely, one defines, given an almost n

complex structure, a function f to be holomorphic if ~f 0 (i.e , df E gl,0); = . one

sees easily, by the local inversion theorem of U. Dini, that the almost complex

structure comes from a complex structure on M if and only if for each p in M

there do exist holomorphic functions F 1 ..... F defined in a neighborhood U of n

p and giving a frame of gl,0 over U. This occurs exactly if and only if the almost

complex structure is integrable: we have thus the following (cf. iN-N] , [H~r] for

a proof).

T h e o r e m 1 . 4 ( N e w l a n d e r - N i r e n b e r g ) o A n a l m o s t c o m p l e x s t r u c t u r e on a C

m a n i f o l d c o m e s f r o m a ( u n i q u e ) c o m p l e x s t r u c t u r e i f a n d on ly i f i t is i n t e g r a b l e .

Following Well ([We], p. 36-37) we shall give a proof in the case where

everything is real-analytic, because then we see why this is an extension of the

theorem of Frobenius that we now recall (see [Spiv I] for more details, or [Hi]).

Theorem 1.5• Let ~PI' .... ~ be l-forms defined in an open set f~ in RR n and r

linearly independent at any point of f2. Then for each point p in Q there do exist

local coordinates x I ..... Xn such that the span of <0 I, .... <0r equals the span of

dx I, .... dXr, <==~> <01 ..... ~r spana differential ideal (i.e., V i = i ..... r 3

forms ~ij (j=l .... r), s.t. d~ i = ~r • j=l ~j A Jij).

Proof. The usual way to prove the theorem is to consider, V pS in f~ the space

V , of tangent vectors killed by ~I ..... q0 : then in a neighborhood U of p there p r

exist vector fields Xr+ l ..... X spanning V t for any p' in U. Since n p

~ i ( [ X j ' X k ] ) = Xj(cPi(Xk) ) - Xk(~i(Xj) )- d,~@i(Xj,Xk)

we see tha t the v e c t o r f ie ld [Xj, Xk] at each pl in U l ies in V p t . One looks then

for coordinates x. i ..... x s°t. V , is spanned by 8/8Xr+ I ..... 8/ax , and these n p n

coordinates are obtained by induction on (n-r). In fact, by taking integral curves of

the vector field X , one can assume X = 8/8x , and replaces X by Y = X. - n n n I I 1

(X.x)X , which span the subspace W ~ of vectors in V ~ killing x , and so z n n p p n

a l s o the v e c t o r f ie ld [ Y . , Y . ] at each point p~ in U l ies in W i (if X(x ) = O, z j p n

Y(Xn) : 0 ~ iX,Y] (Xn) : 0!). By induction there are coordinates (Yl ..... yn ) with

Z n Wpt spanned by 8 / S Y r + i , . . . , O / S y n-1 o We can r e p l a c e Xn = j= t a j (y ) (O /SYj ) by r

Yn = ~ j = l a j ( y ) (8 /Sy j ) +an(Y ) (8 /8yn ) ; s ince [ ( 8 / S g i ) , Y n ] (i = r +1 . . . . . n - i ) equa l s

i 0aj(y) 8 j ~ n+l ..... n- I 8Yi 8yj

but on the other hand, this vector field is in V ~ , thus it is a multiple of Y by a p n

function f. But then, on the one hand, [(8/Syi),Yn ](x n) = 0 (since Yn(Xn) =

X (x) = I!), on the other hand this quantity must equal fY (x) = f. Hence the n n n n

functions a.(y) (j =i ..... r,n) depend only upon the variables Yl ..... Yr'Yn' so, ]

by taking integral curves of the vector field Yn , we can assume Yn = 8/8y n also.

Q.E.D.

We have given a proof of the well known theorem of Frobenius just to notice

that the only fact that is repeatedly used is the following: if X is a non zero vector

field, then there exist coordinates (x I ..... Xn) s° t° X = 8/SXn This follows from

the theorem of existence and unicity for ordinary differential equations and from

Dini's theorem. Both these results hold for holomorphlc functions (they are even

simpler, then), therefore, given a non zero holomorphic vector field Z = n n

~i=l ai (w) 8/8w'z onan open set in ~7 (i°e°, the a 'sl are holomorphic functions),

there exist local holomorphic coordinates z I ..... Zn around each point such that

Z = a / a z n .

The conclusion is that the theorem of Frobenius holds verbatim if we replace

n , and we require local ~n by (17 , we consider holomorphic (I,0) forms ~I .... '~°r

holomorphic coordinates z I .... ,Zn s.t° the ~7-span of ~I .... " ~r be the ~7-span

of dz I ..... dZr. The proof of the Newlander-Nirenberg theorem in the real ana-

lytic case follows then from the following.

Lemma 1.6. Let fi be an open set in ~2n let tUl ..... ~J~n be real analytic com-

plex valued 1-forms defining an integrable almost complex structure (i.e., 1.4

holds). Then, around each point p £ Q, there are complex valued functions

IVl .... ,Fn s.t. the span of dF 1 ..... dFn equals the spanof to I .... ,Wn.

P r o o f ° T a k e l o c a l c o o r d i n a t e s . . . . x a r o u n d p s . t . e a c h ~ is e x p r e s s e d 2n X l ' I~ n c~

by a power series ~j=l ~I< f-',u~j I~ x dx. , where I~ = (k I ..... kgn) denotes a multi- J Zn

index. Then ~ = ~. I~ ~ ~ !~ xl~ dx. and, if we consider ~(Zn as contained in C t J J, J

the monomial x IK by the monomial z IK and x.j by dzj (here x.j is upon replacing

the real part of zj!), t~c~ and t~ extend to holomorphic 1-forms t~c~' ~c~ in a neigh-

borhood of p in ~ Z n S ince ml . . . . ' ~ n ' ~ i . . . . . '~n a r e a l o c a l f r a m e fo r g l

the Wo~'S, ~¢c's give a basis for the module of holomorphic 1-forms, therefore one

can write

By r e s t r i c t i o n to ~ 2 n , u s ing (1o4) we s e e tha t ~ S V =- 0, h e n c e w 1 , . . . , ~ n span

a differential ideal, hence Frobenius applies and there exist new holomorphic coordi-

Zn nares in C , w I .... ,WZn s.t. the span of dw I ..... dWn equals the span of

t~l,...,Wn " We simply take I v .i to be the restriction of w.1 to IR Zn . Q.E.D.

Remark 1.7. Assume that for t = (t I, .... tin) in a neighborhood of the origin in

Cm one is given real analytic l-forms ~Jt, l ..... oJt, n as in lemma Io6 which are

expressed by convergent power series in t I ..... tin, and define an integrable almost

complex structure when t belongs to a complex analytic subspace B containing flhe

origin. Then, for t in B, the conclusions oflemma 1. 6 hold with Ft, I ..... Ft, n

expressed as convergent power series in (t I ..... tin). In fact, if a vector field X t

is given by a convergent power series in t I ..... t also the solutions of the asso- m

ciated differential equation are power series in t I ..... t : moreover, by the local m

inversion theorem for holomorphic functions, if f(x, t): ~/ -~ Q is locally invertible,

real analytic in x and complex analytic in t, then the local inverse is also complex

analytic in t.

§ g. Small deformations oi! a complex structure

If U is a vector subspace of a vector space V, and ~V is a supplementary

subspace of U in V (thus we identify V with U e W), then all the subspaces U',

of the same dimension, sufficiently close to U, can be viewed as graphs of a linear

m a p f r o m U to W: we a p p l y t h i s p r i n c i p l e p o i n t w i s e to d e f i n e a s m a l l v a r i a t i o n o f

a n a l m o s t c o m p l e x s t r u c t u r e ( h e n c e a l s o of a c o m p l e x s t r u c t u r e ) °

Definition 2. I.

T 1,0 ® (T O , l)V

A small variation of an almost complex structure is a section ip of

(the variation is said to be of class C r if ip is of class cr).

Remark 2.2. To a small variation ~ we associate the new almost complex struc-

ture s.t° T O , 1 = [(u,v) ~ T I'0 ~9 T O, 1 [ u =~(v)} , since there is a canonical iso-

ip TI,0 m o r p h i s m o f ® (T O , 1)V w i t h H o m ( T 0, 1 , T t , 0 ) .

We assume from now on that M is a complex manifold:

h o l o m o r p h i c c o o r d i n a t e s (z 1 . . . . . Zn } o n e c a n w r i t e ip a s

(2 .3 ) ip =

(I 8z

then, in terms of local

so that

: , U <Z : <00; v 8z 8 8

andis annihilated by ( T I ' 0 ) V , the span of [m : dz - ~$ <gg dE B ] C~ O~ (~, "

' s , w h e r e Y

On the other 0 , 1

h a n d , b y w h a t w e ' v e s e e n Tip i s s p a n n e d by t h e

g v og ¢ oz ~{ ct c~

Since dw = -~ dip ~ A dE , we are going to write down the integrability condition

(1.4), which can be interpreted as

( 2 . 4 ) dmc~(~ {, ~6) = 0 V a., 4{, 8 ( ¥ < 8) .

W e h a v e

-dw(l = ~.~ \-~z dz A dK 8 + ~ dE¢ A dE8

which belongs to gl, I (9 gO, 2 , hence kills pairs of vectors of type (I,0). We get

thus the condition

- - ' ipct' Oz ' 8g 8 y &

+ dw e O , ip(~" = 0 ,

boiling down to

( 2 . 5 ' ) ~co ~ ~o 7 ~o ~

8~ 8~--~- + 8z ¢ ? ¢ ¢

a~ 7 c~

8z ¢

- - %o = 0. ¢

The condition that (2.5') holds for each c~

a s

(2.5)

w h e r e

C~ y < $

1

, a n d y < 6 , c a n b e w r i t t e n m o r e s i m p l y

t d~ A ® O_J___ Y 3zc~

7

0z¢ ¢ - Oz We y a T

, , (X 6

We s h a l l e x p l a i n t h e s e d e f i n i t i o n s w h i l e r e c a l l i n g s o m e s t a n d a r d f a c t s o n D o l b e a u l t

c o h o m o l o g y a n d H o d g e t h e o r y ( h a r m o n i c f o r m s ) .

So, l e t V be a h o l o m o r p h i c v e c t o r b u n d l e , a n d l e t ( U ) be a c o v e r of M b y

------ U × ~7 r h e n c e f i b r e v e c t o r c o - t open sets where one has a trivialization V IUc~ c~

ordinates vc~ , related by vcc = g(x~ v~ where gc~ is an invertible r X r matrix

of holomorphie functions. We let g0, P(V) be the space of (C ~ ) sections of

V ® AP(T 0, I)V: since 8 gccB = 0, it makes sense to take 8 of (0, p) forms with

values in V (i. e. , elements of gO, P(V)), and we have the Dolbeault exact sequence

of sheaves

E1 ~0, 5z 5 0 ~ ~ ( v ) - ~ ~ ( v ) , l ( v ) • . . ° n~0, n(v )_~ 0 ,

where @(V) is the sheaf of holomorphic sections of V. We have the theorem of

Dolbeault (the g0, k(v ) are soft sheaves)°

Theorem 2.6.

ker H°(0 i + 1 ) HI(M, ~ (V))

Im H°(8.) I

So ~ is well defined for our ~0 E ~0, I(TI,0). For further use, we shall use the

notation ® = @(T I'0). To explain the bracket operation, we notice that this is a

h i l l n e a r o p e r a t i o n

10

[ , ] : go, p ( T l , O ) X go, q ( T l , O ) -4 go, p+q(T 1 ,0)

w h i c h in l o c a l c o o r d i n a t e s (z 1 . . . . . Zn )' if

and

=

I= [i I <--o o<i P

- f 7 ] ° " ° i a z ~ az

p ~ I, O~

J d A7 a = g~ ® a-7- '

J , ¢ ¢

is such that

[~ 0] = I d~ AI A d~ AJ ~ f[ 8g¢ 8 ~ 8 ' ct ~ az - g~ 8z 8z

l,J,~,¢ ¢ ¢ &

The bracket operation enjoys the following properties

i) [q~,~] = ( - I ) p q + l [o,qJ]

( 2 . 7 ) ii) ~[t0,@] =[~,~] +(-])P[,O,~¢]

iii) if ~ is in go, r(Tl,O), then the Jacobi identity holds, i.e.,

( - 1 ) P r [ ~ , [ O , ~ ] ] + ( - 1 ) q P [ ~ , [ ~ , t o ] ] + ( - 1 ) r q [ ~ , [~o ,¢] ] = 0 o

Before recalling the Hodge theory of harmonic forms, we remark that, if we

have a small variation O(t) of complex structure depending on a parameter

t = (t I ..... tm )' setting B = it I ~O(t) =-~ [o(t),O(t)]], ]B is precisely the set of

points t for which O(t) defines a complex structure: but in order that the complex

charts depend holomorphically upon t for t in B (we assume, of course, that

~0(t) be a power series in t I ..... t ), we want (cf. remark Io6) B to be a complex m

subspace. The l~uranishi family, as will be explained in the second lecture, is a

natural choice to embody all the small variations of complex structures with the

smallest number of parameters.

Now, let V be again a holomorphic vector bundle on M, and assume that we

choose Hermitian metrics for V and TI,0 so that for all the bundles V® (T °'p)V

is determined a Hermitianmetric (if M is C ~, we can assume the metric to be

C~). Thus a volume form d~ is given also on M, and thus, for ~0,0 E go, P(V)

a Hermitian scalar product is defined by (t9,@) =L(~,@> du. ((:9,qa) is the ~M X X

value which the Hermitian product, given for the fibre of V® (T °'p)V at the point x,

takes on the values of ~ and @ at x)°

11

It is t h e r e f o r e d e f i n e d the a d j o i n t o p e r a t o r 8": 8 ° ' p + I ( V ) -~ 8 ° ' p ( V ) by the

u s u a l f o r m u l a ( 8 ~ , g ) = (M, 8 g), a n d one f o r m s the L a p l a c e o p e r a t o r

- - - - ' l ~ - ~ I ~ -

S = 8 8 + 0 O .

C o , C ° , We have [] : P(V) -* P(V) and the space of harmonic forms is

(z.8) ~ P ( v ) = { e s e ° ' P ( v ) I D ~ : 0 ] = { e l a e : g*¢ = 0 ]

The m a i n r e s u l t i s t h a t one h a s a n o r t h o g o n a l d i r e c t s u m d e c o m p o s i t i o n ( w h e r e we

s i m p l y w r i t e 8 P f o r 8 ° ' P ( v ) )

(2..9) ep = 3£ p ~ ggp-1 ~ 5* g p+I

R e m a r k 2. 10. 3£ p ~ ~ 8 p - 1 c o n s i s t s of the s p a c e F (ke r 8)

p - f o r m s : in f a c t if 0 O M = 0, t h e n 0 = ( 3 8 M,M) = II II

i n v i e w of £ ) o l b e a u l t ' s t h e o r e m one h a s the f o l I o w i n g

of all the ~ closed

-----> 8 %0 = O. Therefore,

T h e o r e m 2. 11 (Hodge) . ~cP(v) is n a t u r a l l y i s o m o r p h i c to HP(M, (9(V)).

o v e r f o r e a c h M E 8 ° ' p ( V ) t h e r e is a u n i q u e d e c o m p o s i t i o n

q) = ~? + [] qa , w i t h ~? = H(c?) E 3{ p , ~ : G(eo) E (3{P) ±

M o r e -

H is o b v i o u s l y a p r o j e c t o r ( the " h a r m o n i c p r o j e c t o r " ) on to t he f i n i t e d i m e n s i o n a l

s p a c e 3£ p , w h e r e a s G is c a l l e d the G r e e n o p e r a t o r . We r e f e r to [ K - M ] a g a i n

f o r t he p r o o f of t he f o l i o w i n g

- -;~ - --I-"

Proposit ion Z. 1Z. 8, O c o m m u t e w i t h G, a n d the p r o d u c t of 0, O o r G w i t h H

on b o t h s i d e s g i v e s z e r o .

§ 3. H u r a n i s h i ' s e q u a t i o n a n d t he I C u r a n i s h i f a m i l y

F i x o n c e f o r a l l a n H e r m i t i a n m e t r i c on T 1 , 0 a n d l e t ~ P be 5{p(T 1, 0): we

c a n t h e r e f o r e i d e n t i f y , by H o d g e ' s t h e o r e m , h a r m o n i c f o r m s in ~ P w i t h c o h o m o I o g y

c l a s s e s in H P ( M , ® ) . R e c a l l a l s o t h a t , by ( 2 o 7 . i i ) , a b r a c k e t o p e r a t i o n is d e f i n e d

[ ] : H P ( M , ~ ) × H q ( M , ® ) -, H P + q ( M , ® ) .

Let ~i' ~]m be abasis for 5£1 with . . . . . so that we can identify a point t E m m

the harmonic form ~i= 1 t.1 ~3.i " Consider the following equation

( 3 . 1 ) ~ (t) = ti ~ i + ~ G [ ~ ( t ) , ~ ( t ) }

It is easy to see that one has a formal power series solution M = ~m=l C~m(t) '

12

where %0 (t) is homogeneous of degree m in t: in fact by linearity on t of m

O , G

o l ( t ) = t i n i , ~ z ( t ) = ~ , = . . . . .

The power series converges in a neighborhood of the origin because G is a regular-

izing operator of order Z (with respect to H~Ider or Sobolev norms).

We want to show that ]5 = [ t I ~(t) converges, and defines a complex

structure on M} is a complex subspace around the origin in fm . We know that

]B = [ t I ~%0(t) - --~ [~0(t), <P(t)] = 0 } and we claim that the follo%,ing holds

/,emma 3. Z. ~(t) - ½ [~(t),~0(t)] = 0 if and only if H[~p(t),~(t)] = 0

Proof. The "only if" part is clear, since H~ = 0. Conversely, we want to show

that d~ = ~- -~ [<0,~0] equals zero. Now ~0 ='~l + ~ ~*G[cp,<0] by ICuranishi's

equation, and ~01 is harmonic: hence

I 1

_ - f , ~ _ ~ ' , < _

But the identity id equals H + [3G : }{ + 88 G + 3 3G, thus (since H[~,~] : 0 by

assumption) -Z$ = a 8G[<0,$] = (since 0, G commute)= O G ~[%9,¢9] = (by Z.7)

= 28 G [~,~] = (since [[©,~],-"0] = 0 by Jacobi's identity)= 2~*G[~,%0] . We

have therefore reached the conclusion that ~(t) = -~*G[~(t), <0(t)] , in particular

for anySobolevnorm II II, ll~(t)ll ~ cost ll+(t)ll ll~(t) ll But since ll~(t) ll is

infinitesimal as t -~ 0, we get that for t small llqb(t)[l < ll%b(t) ll, hence l[~(t)ll = 0

and ~(t) = 0 as we want to show. Q.E.D.

We use now the standard notation hi(~) = dim~ Hi(M, ~), for a coherent

sheaf on M, and we state an immediate consequence of (3.2).

• h 1 Corollary 3 3. If m = (®) as before, k = hZ(®), then B is defined by k holo-

morphic functions gl ..... gk of t = (t I ..... tm) which have multiplic{ty at least 2

at the origin. Moreover, if we identify C m with HI(®), •k with H2(®), the func-

tion g(2): cm -~ ~k given by the quadratic terms of gl .... 'gk corresponds to the

quadratic function associated to the symn~etric bilinear function

[ , ] : HI(®) X HI(®) -~ H Z ( 9 ) .

Proof. Let ~I ..... {k be an orthonormal basis for ~2 . If u~ is in g0'Z(Tl'0),

H•= 0 is equivalent to {~, ~.) = 0 for i = I ..... k° Therefore, by lemma 3. Z, B I

is defined by the k functions gi(t) = ([Q) (t), <p (t)] , ~ l) = 0. Since

13

~(t) = 52 t.l ~]l" + o(t), gi(t), which is clearly a convergent power series in t, has a

McZaurin e x p a n s i o n

m

gi(t) = j,~k=l([~J'~k]' gi)tjtk +o(t2) . Q.E.D.

The final step i s to observe that the varying complex structures on M, parametrized

by t ~ B, can be put together to give a structure of complex space to the product

MXB, we have

Theorem 3.4. On M×B there exists a structure of complex space Z such that the

projection on the second factor induces a holomorphic map H : Z -~ B such that

i) each fiber X t = ~-l(t) is the complex manifold obtained by endowing M

with the complex structure defined by ~)(t);

ii) for each point p E M = X 0 there exists a neighborhood U in M and a

neighborhood V in % such that V is biholomorphic to U X IB under a

map ~: U X B -* V s.t. rro ~ is projection on the second factor.

Sketch of Proof. By remark 1.7, for each point p in M there is a neighborhood

U and functions Ft, i(x) (i-l .... ,n) s.t. for any t in B they give a local chart

for the complex structure defined by ~(t). Let F(t,x) = (Pt, 1 (x) ..... Ft, n (x)):

17 -~ ~n ; we use (F,t): U X B -~ C n X D to give the local charts for the complex

structure %. The inversion theorem of U. Dini ensures then that the complex

structure on % is globally well defined, and that ii) holds. Q.E.D.

14

LECTURE TWO: DEFORMATIONS OF COMPLEX STRUCTURES AND KURANISHI'S THEOREM

In this lecture lwill review the notion of deformation of complex structure

introduced by l~odaira and Spencer, the notion of pull-back, versal family .....

define the l~odaira-Spencer map, and state the theorem about the semi-universality

of the I~uranishi family.

§4. Deformations of complex structure

Let M be as usual a compact complex manifold.

Definition 4. i. A deformation of M consists of the following data: a morphism

of complex spaces [I: Z -~ B, a point 0 E B, an isomorphism of the fiber X 0 =

I]-1(0) with M s.to H is proper and flat.

- l (b ) R e m a r k 4 . 2 . ti i s s a i d to b e s m o o t h i f V b ~ I% t h e f i b r e X b = [[ i s s m o o t h

( and r e d u c e d , o f c o u r s e ] ) . A d e f o r m a t i o n [I is s m o o t h (at l e a s t i f o n e s h r i n k s B)

b y v i r t u e o f t h e f o l l o w i n g

Lemma 4.3. Let I] : C3B, = A -~ (9%, = R be a homomorphism of local rings of o p

complex spaces, and assume that 2;'~ makes R a flat A-module, and that more-

over 11 has a smooth fibre, i.e. R/~q ~A =~ ~ [~i ..... ~n ] ' ?~A being the maxl-

real ideal of A. T h e n R -~ A[x l,...,x n ] .

Proof. Let x I ..... x be such that x. maps to ~. through the surjection R -~ -'i< n I i

R/II;I~'~A . Thus [[ defines a homomorphism f: A ix I ..... Xn] to R. f i s surjec-

tire by Nakayama's lemma, and we claim that flatness implies the injectivity of f.

Let I< = ker f, so that we have an exact sequence

0 ~ t~ -~ A [ x 1 . . . . . X n ] ~ R ~ 0 .

T e n s o r i n g w i t h t h e A m o d u l e A/7~ A -~ • we g e t , s i n c e T o r l ( R , A / ~ A) = 0 ( s e e

[ D o u 2 ] , p r o p o s i t i o n 3) a n e x a c t s e q u e n c e

0 '~ K ® A / ~ A ~ ~E {E 1 . . . . . ~ ] < R/n$7~A "* 0 , n

S i n c e E i s , b y a s s u x n p t i o n , a n i s o m o r p h i s m , N ® A / ~ A = 0, h e n c e K = 0, a g a i n

b y N a k a y a m a ' s I e m m a ( a p p l i e d t o K a s a n R m o d u l e ! ) . Q.E.D.

15

Remark 4.4. A deformation is said to be smooth if B is smooth: in this case ::

is just a proper map with surjective differential at each point. Lemma 4.3 shows

that property ii) of theorem 3.4 holds for every deformation. In the case when B

is smooth, a classical theorem of Ehresmann ([Eh]) asserts that :: is a differ-

entiable fibre bundle. This is a local result, and to give an idea of the proof

(especially to stress the importance of vector fields:), we can assume 13 to be a k

cube in ~k . Then one wants to show that % is diffeomorphic to MXB visa map

compatible with the two projections on B: one assumes ~ to have a Riemannian

metric, so that, if x I ,x k are coordinates in [Rk . . . . . one can lift the vector field

O/0x k to ~ to a unique vector field ~ that is orthogonal to the fibres of :. Then

one proves the result by induction on k: if I~ ~ = 13 N IR k-l, one takes the integral

curves of ~ to construct a diffeomorphism of

applies induction (FI-I(B l) ~ M× ]:% l) to infer

Riemannian metric included) is C , the above

An analogous result holds in the general case°

::-I(BI)X (-I, I) with %, and then

that ~ MXB. If everything (the %U

proof yields a C diffeomorphism°

Theorem 4.5. Given a deformation ::: ~-~ 13 (shrinking ]3 if necessary) there

exists a real analytic (C °~) diffeomorphism V: M XB -+ • with ::o y = projection

of M×B -~ B, and such that y is holomorphic in the second set of variables.

Idea of proofs. In the C c° case (cf. [Ifu 3] , p. 19-23) the proof is easier: there

exists a finite cover V of % such that, if Ua = M f: V (M is identified with X ), (% (% o

V ~- U ×13 under a biholomorphism ~0 . We can assume % ~ ~N: using these c~ cc ¢%

co

Cp's and a partition of unity subordinate to the cover V we can define a C rnorph- CC C~

ism of % to a tubular neighborhood T of M in IN , and then we compose with a

retraction of T to M to get ~0: ~ -~ M such that ~011vl = identity. Then ~ X

gives the required diffeomorphlsm.

In the real analytic case, one can use the fact that, if TN{ is the real tangent

bundle of I%~, HI(M, TM) = 0: then the power series method of [K-N~I], pp. 45-55

gives the desired result°

Using the diffeomorphism V: M×B -~ ~, for each b E ]B one gets a small

variation of complex structure %0(b) ~ A 0' I(TI'0) which depends holomorphically

upon b. If (B,o)c (~r,o) and t l,...,tr are coordinates on ~r (r = dim ~B,o /

Y~B,o2 ) one can write ~(b) = ~ t i H i + oct), and the linear map 0 from ~r = TB,o

(Zariski tangent space to B at 0) to HI(®) such that -O (8/8t i) = class of T] i in

HI(®M), is called the I~odaira-Spencer map, and we shall soon give an easier way

to define and compute it.

16

Definition 4.6. Let lh ~-~ B be a deformation of M, and let V be a (finite) cover cx

of Z w h i c h l o c a l l y t r i v i a l i z e s I2, i . e . s u c h t h a t t h e r e e x i s t s a b i h o l o m o r p h i s r n -1

cp : VC~-~ UC~×B (Uc~ = VC~ NM, and we tacitly assume II o ~c~ to be the projection

of U × B -~ B). Let ~ be a tangent vector to B at 0, and let t9 be the unique

l i f t i n g of g ( v i e w e d as a c o n s t a n t v e c t o r f i e l d in C r ~ 13) to V g i v e n by the t r i v i - (I

a l i z a t i o n ~P~. T h e n ~ c t - "~B' r e s t r i c t e d to Uccf~ U ~ , i s a v e r t i c a l v e c t o r f i e l d ,

t h u s ( v ~ - v~)E HI (M , ®). We d e f i n e P" T B , o ~ H I ( ~ ) to b e t he t [ n e a r m a p s u c h

t h a t p ( g ) = va0c- ~ a n d i t i s e a s y to s e e t h a t p is w e l l d e f i n e d , i n d e p e n d e n t l y of

t he c h o i c e of the c o v e r a n d of the t r i v i a l i z a t i o n s ( for i n s t a n c e , c h a n g i n g t r i v i a l i z a ~

t i o n s , ~ i s r e p l a c e d by v ~ ' s u c h t h a t d - v a ' is a v e r t i c a l v e c t o r f i e l d , t h e r e f o r e

~j. - v~8 i s c o h o m o l o g o u s to ~'Cc - ~ ) " p i s c a l l e d t he 1 4 o d a i r a - S p e n c e r . m a p °

T h e N o d a i r a - S p e n c e r m a p g i v e s the f i r s t o r d e r o b s t r u c t i o n to the g l o b a l l i f t -

a b i l i t y of a v e c t o r f i e l d , a n d i t s i m p o r t a n c e l i e s in i t s f u n c t o r i a l n a t u r e , t h a t we a r e

now going to explain.

Definition 4.7. Let I~: ~ -~ B be a deformation of M, and let f: B' -+ B be a

morphism of complex spaces, ©' a point of ]3' with f(O') = 0. Thenthe pullback #

f (~) is givenby ~l = {(x,b,)ix E ~, b' E B' s.t. l](x) = f(b')~ C~X B', with

[[' induced by projection on the second factor.

Let f~:* TB',o~ -~ TB,o be the differential of f at O' , andlet p, p' be

the respective 14odaira-Spencer maps of Z, ~': we have

(4.8) 0 ~ = p o f~

as it is immediately verified.

Grothendieck's point of view was, in particular, that in order to compute

0(~) it suffices to choose ]3' = ~t E ~E I t2 = 0] , and f the unique morphism of

B'-~B sot. f,(8/St) = ~ ~, also, in the context of pull-back, the meaning of p= 0 is 2 l~I

that, if B I is the subspace of 13 definedby ~B,o ' i: -0 B is the inclusion,

then i (~) ~- B I× M if and only if 0 = 0.

W'e can thus verify that the two definitions we have given of P do, in fact,

coincide, limiting ourselves to 1-parameter deformations.

Lemma 4.9. Let If: Z-~ B be a deformation of M with base B = it E~EIt 2 = 0] ,

whose associated small variation of complex structure is given by the form ~0(t) =t~,

with ~ E A 0'l(Tl'0): then, using the Dolbeault isomorphism, p(8/at) is the class

of ~ in HI(®).

17

C~ Ct Proof. We choose trivializing charts on UCX 13, with z I ..... z n

UC~, g i v e n b y ~ ( z , t ) = z . + t w . . ~ ( t ) o n U i s e x p r e s s e d b y J J ~

t E q~ d~'ct ® ~ i,j J 8z?

1

and the condition that ; ? be holomorphic is that (g +~)({?) = O, i.e. 1 I

- o~ j _~ 8 w . + q d z . = 0 ,

l • J J

coordinates i n

hence locally ~1 can be expressed as

E ~ ~ a w .

i 1 Oz~ 1

In view of the way the Dolbeault isomorphism is gotten, it suffices to verify that, if

we set

E c~ - w i ® = A

i az. ~ 1

then A s - A8 is cohomologous to ~9c~ - ~@. Now @C~ = 8/8t in the <~-chart, there-

fore, expressing 0 - ~)~ i n t h e C o - c h a r t , we g e t , i f ~ = f ~ ' ~ ( ~ B , t ) , i s t h e c h a n g e J J

o f c o o r d i n a t e s ,

8f[ ~B(~8 t).t E ' a ~ ~(z B a

- - gi ) i at = 0 8z. a i az. 1 I

if we set

C~ f~B(¢$,t) = h.~B(~)+tgjj ~(~8) .

But this last expression equals ~ c~ • = z. + tw. ~, hence, by the chain rule, J J J

and we are done, since

wi =g B k @z k

ah ~B 8 = E t 8 C~

az~ i ~Zk8 ~z i

b . Wk

Q.E.D.

18

§ 5. lCuranishi's theorem

Definition 5o 1. A deformation II: % -~ B of M is said to be corr~plete if for any

other deformation ~ql Z' B' : -~ , there exists a neighborhood 13 ~' of O t in ]B t and

f: B t~ -~ B such that Z ~ = Z' IB ~' is isomorphic to the pull-back f (Z). The de-

formation ~: Z-~B is saidtobe universal if it is complete, and moreover f is

(locally) unique (respectively: semi-universal if it is complete and

f~:'~ TB', o' -~ TB, o is unique).

Remark 5o Zo In view of (4.8), a complete deformation is semi-universal if the

associated I~[odaira-Spencer map p is injective° Let us see that 0 is surjective

for a complete family; in fact, by lemma (4.9) and the }<uranishi equation (3. I), the

Kuranishi family has a bijective I~odaira-Spencer map: hence, if a complete family

exists, it must have a surjective Kodaira-Spencer map.

Proposition 5o3. A semi-universal family is unique up to isomorphism.

Proof. By completeness, if -q: %-~IB , ~q': %' -~ B' are semi-universal, there

exist f': B-~B', f: B'~ B with %'= f;'~(%), Z = f'"(Z'). Hence %= (f'f)~(%) ,

and by semi-universality (f ).,~ identity, but also (f f ).,~ = id, therefore, by the

local inversion theorem, f, f' are isomorphisms. Q.E.D.

We can now state the theorem of I<uranishi, referring the reader, for a com-

plete proof, to [Dou I], [I~u 3], [I~u g], and to [I<-M] and [Ku I] for weaker versions.

Theorem 5.4 (ICuranishi)

i) The I~uranishi family is semi-universal, and f~ coincides (up to sign)

with the ICodaira-Spencer map p o

ii) The lCuranishi family is complete for b E B - to ] , when viewed as a

deformation of X b .

iii) If H°(®M) = 0, the I<uranishi family is universal.

Let's draw some corollaries of the above theorem, noting that by proposition 5.3,

the lluranishi family is "the" seml-universal family of deformations, and that, by

(3.3), the I<uranishi family is smooth if H2(®) = 0°

Corollary 505° Ifa deformation If': Z'-~B' has a smoothbase B', and surjective

Kodaira-Spencer map p' , then it is complete and moreover the I<uranishi family of

deformations is smooth.

Ig

Proof. Let f : B' -~ B be such that %' : f~:'~(%). By taking a smooth subn~anifold of

B' we can assume f, to be bijeetive° But then B' is a neighborhood of O E C r

and we have f ~r r -~ B c with f:~ invertible: by the local inversion theorem

f((~r) contains a neighbourhood of O E r , hence gives a local isomorphism

f: B ' -~ C r ~ IB ~ C r Q.E.D.

Finally, we mention a refinement of part i[i) of the theorem ([Wav]).

Theorem 5.6 (-VVavrlck). If the base B of the Kuranishi family is reduced,and

h°(Xt , ®t) is constant, then the Kuranishi family is universal.

In the next paragraph we shall discuss the example of the Segre-Hirzebruch sur-

faces, which illustrates how certain statements in the above theorems cannot be im-

proved. We only rernarkhere that if M is a curve (n:l), the HZ(®) =0 and the

Kuranishi family is smooth of dimension hi(®) = 3g- 3 + a, where g is the genus

of the curve and a = h°(®) is the dimension of the group of automorphisms of M.

§6. The example of Segre-Hirzebruch surfaces

The Segre-Hirzebruch surface ~ (where n E IN) is, in fancy language, the n

1 b u n d l e IP(Vn) a s s o c i a t e d to the rank. 2 v e c t o r b u n d l e Vn such that O{V n) ~

--~ (%pi (9 (% l (n) . By a b u s e of l a n g u a g e we sha l l i den t i fy V wi th (%(V), t h e r e f o r e we

ge t a s p l i t e x a c t s e q u e n c e

(6. i) O -~ (9 1 -~ V -~ (% l(n) -~ 0 . n

P

We consider all the rank Z vector bundles V which fit into an exact squence like

(6. I), which are classified by HI((% l(-n)), a vector space of dimension (n-l), and

we consider the family of ruled surfaces [P(V), thus obtained, as a deformation of n-1

IF n . In concrete terms, we take B = • , with coordinates tl, .... in_l, and we

obtain ~ glueing ~ °l X CX B (= [ °l x (~ °I- [~o])×B) with ~ °l x CX B (= ~ °l x

l I (~ ol - [o})X B)bythe identification of (y0,Yl, Z,tl ..... tn, I) with (y0,Yl, z ,

t I ..... tn_ I)

n-i

i , -n t -i

(6, Z) z'= --z ' Yl = Yl z " Y0 = Y0 + Yl tlz i=I

! l ° (note t h e n tha t YO = YO ~ Yl ~ i t i z n - i )"

Now we sha l l c o m p u t e the K o d a i r a - S p e n c e r m a p of the f a m i l y [I: % * B we

have just constructed.

20

t

(f) ( f ) ( f ) 1 1 1

where the prime means w e are writing a vector field using the second chart. So,

expressing (8/0ti) using the first chart, we get

( 8 ) , zn-i) O " 8 P ~ = -(Yl -- = (-Yl z-l) x OY0 8Y0

We shall now show that these vector fields generate the Cech cohornology group

HI([u,u'}, ®~ ), where U = [pl x C of the second chart, amdwe notice that, since

HI ( @R ml x n ) = 0 , the Cech cohornology group we are going to compute is indeed

Hl(~ n, ®Fn), hence the Kodaira-Spencer map will be bijective, and the constructed

family will be the Kuranishi family. , -n , [pm

Let's w o r k on ~n , where z'= I/z, Yl = YlZ ' Y0 = Y0: since on

we have the E~/ler exact sequence

0) ' ' ' ° ' ~ X

(6.3) 0 -~ @~m @prn(1)rn+l • rn ®~rn ~ 0 x r n Ox 0

)

@ 8 , El 8 Therefore the vector fields in lP 1 have as basis x 0 @x0 , x 0 ~x I ~x 0 .

holornorphic sections of ®~. on U N U' can be written uniquely as n

(6.4) ! zi aio0 Y0 -- + Y0 -- + all0 Yl -- + bj -- i 6 ~ Oy 0 ai01 8y I 8y 0 j ~ ~ 8z

where ai00 , ai01 , all 0, b. £ C. These sections are holornorphic also on U if J

only non zero terms occur with i E EN, j E iN. Since

8 0

OY o

8 2 O 8 8 3 n , 8z t = -z ~z + nzy 0 , , - -- z ,

8Y 0 8Y I 8Y 1

we write a regular section on U' in terms of the first coordinate, and we have

(z°) I ai00Y 0 "/"7+8y 0 ai0 ly + all0 Yl ~ + 8t iE~N jE[~

0 0 Z-i 0 8 zn-i+ t ~ z -n-i ~ b -- z "j+z = ~_~ ai00Y 0 + ai01Y 0 ail0Yl - i E~ °Yo j E~ j o~z

j E ~

21

Since HI(®~ ) n

= H°(U n U', ®~. )/H°(U, ®~ ) + H°(U ' , ®~ ), we see that n n n

-1 8 - ( n - l ) 8 } HI(®07 (6.5) Yl z -- , .... yl z -- is abasis of ) . 8Yo 8Yo n

H° (~n o F u r t h e r m o r e , s i n c e , ®~ ) = H (U, ®~ ) [~ H°(U ' , ®~ ), we have t ha t n n n

8 8 n D 8 8 2 8 8 (6.6) Yo ' Y o - - ' . . . . z Yo ' a ~ ' ~ ~ ' z ~ - n Y o Z

8y 0 8y I 8y 1 8Yo

(and Yl (818Y0) if n = 0) are a basis of H°(~n, ®~ ). n

Corollary 6.7. hl(®~.n) = n-l, h°(Q~n ) : n+5 if n > 0, 6 if n=0, hence

~" -~ ~" if and only if n = m. Furthermore, the family defined through the glueing n m

(6.2) is the Kuranishi family of [~ n

Let now T k c B be the determLnantal locus

(6.8) T k = t I rank (tl 1) 1 t2 tk+2 < k

tn-k- 1 tn- 1

We r e f e r to ([Ca 1], §1) for the p ro o f of the fo l lowing

P r o p o s i t i o n 6 . 9 . T k is an a l g e b r a i c cone of d i m e n s i o n m i n ( Z k , n - 1 ) , and if

t 6 T k - T k _ l ' t h e n Xt ~- ~ 'n-Zk "

R e m a r k 6. 10. Th i s e x a m p l e i l l u s t r a t e s how the K u r a n i s h i f a m i l y can be s e m i -

u n i v e r s a l on ly fo r t = 0 , and c o m p l e t e for t ~ 0 . I n t h i s c a s e h l ( @ x t ) ha s a s t r i c t

m a x i m u m fo r t = 0, and we n o t i c e tha t

(6. i I) h l ( ® y ) is an u p p e r s e m i c o n t i n u o u s f u n c t i o n in t , in g e n e r a l , " ' t 1

and if h (®Xt) is c o n s t a n t on ]B fo r the K u r a n i s h i f a m i l y ,

t hen the K u r a n i s h i f a m i l y (cf. 5 .2) is a l so s e m i - u n i v e r s a l

fo r t ~ 0 .

In th is c a s e , as we h a v e s e e n , h° (®Xt ) is n o t c o n s t a n t : th i s w a s , in W a v r i k ' s

theorem (5.6), a sufficient condition for the universality of the Kuranishi family.

We are going to show that for the above surfaces the Kuranishi family is not

universal.

22

]Example 6. 12. Take n = 2 and our given family %, obtained by the glueing , , = -2 , -i

z = I/Z, Yl Yl z ' Y0 = Y0 i YltlZ . Consider the local biholomorphisrn of

B to B sending t I to t I f (tl) , where f(tl) is a holomorphic function with f(0) = i. -2

Then the pull-back is the family %' given by the glueing ~ i = 1/~ , 11 1 11 l

, = t l f ( t l ) C - I 110 ~0 + q l . But we obtain an isomorphism of ~' with ~, compat-

= I ible with the projections [I, HI on B , ifwe set ~ = z, ~'= zS, y 0 ~0 , y0= T]0 ,

J Yl = 111f(tl )' Y~ = T]I f(tl)" The condition f(0) = 1 ensures that the given isomorph-

ism of [Fn with the central fibre X 0 has not been changed.

23

LECTURE THREE: VARIATIONS ON THE THEME OF DEFORMATIONS

§7 . D e f o r m a t i o n o f a u t o m o r p h i s m s

A s a n a p p l i c a t i o n o f t h e t h e o r e m o f K u r a n i s h i , l e t ' s a s s u m e t h a t G i s a f i n i t e

( o r c o m p a c t ) g r o u p o f b i h o l o m o r p h i s m s o f M . T h e n we c l e a r l y h a v e a n a t u r a l a c t i o n

o f G o n M × B , w h e r e G a c t s t r i v i a l l y o n t h e s e c o n d f a c t o r , t h e b a s e o f t h e

K u r a n i s h i f a m i l y .

If C~ E G, t E B , c l e a r l y cy i s h o l o m o r p h i c o n X t i f a n d o n l y i f <Y'-',*T0't 1 =

T 0'I i. eo 4=> cy..c0(t) = %0(t). Therefore Gc Aut(Xt) for the set I~ G

[t I~. %0(t) = c0(t)] (note that it makes sense to talk of cir. since cT is an automorph-

ism of M = X ). This set is not so weird in general, since cy. ¢0(t) - cp(t) is a power

series in t, we can only say that it is a complex subspace if G is compact, since,

1,0 by integration with respect to the invariant measure of G , we can assume T M to

be endowed with a G-invariant Hermitian metric. For G ~ cT, ~ being holomorphic

on M, ~7~ commutes with 8, but now the G-invariance of the metric implies that

cy~ also commutes with 8 , [] , G, H.

Now G acts naturally on the cohomology groups Hq(®), and we shall write,

as customary,

Hq(®) G = [ q [ ~ H q ( ® ) l C~,,, V = ~ V c~ ~ G ] .

I f w e i d e n t i f y B w i t h a c o m p l e x s u b s p a c e o f H I ( ® ) , w e h a v e

(7. 1) I t E B ICY 1 X t i s h o l o m o r p h i c V c~E G ] = ~ G = IB N H I ( ® ) G

In fact, since ¢p(t) = t + } 8~'~G [¢Ptt), cp(t)] ,

= B G

1 0 G[~®(t) , v~( t ) ] CY~ ~0(t) = ~ t + ~ ~ ~- ,

t h e r e f o r e cyi ¢P(t) s o l v e s t h e I K u r a n i s h i e q u a t i o n f o r cy;t , a n d ~,~,..~(t) = ~(c~,..,<t). T h u s

~mcP(t) = ~0(t) i f a n d o n l y i f t = cyst, a s w e h a d to s h o w °

B u t w e c a n b e m o r e p r e c i s e , b e c a u s e w e h a v e t h a t

B G = { t ~ H l ( ® ) IV ~ ~ G t = ~ o t H[~(t),~(t)] =0 ] but

~ H [ ~ ( t ) , ~ ( t ) ] = H [ m ( ~ t ) , ~ ) ( ~ , t ) ] : ( i f t : ~o t ) = H [ ~ ( t ) , O(t) ]

t h e r e f o r e we g e t t h a t B G i s a c o m p l e x s u b s p a c e o f H I ( ® ) G d e f i n e d b y hZ(®) G

equations of multiplicity at least 2 and such that their quadratic parts are associated

to the symmetric bilinear mapping

24

[ , ]: Hl(®)G× Hl(q) G -~ H2(9) G "

G Z We get thus a lower bound for dim B , and we observe that the family

[I-I(B G) has an action of G which is holomorphic, fibre preserving, and IBG s u c h

t h a t t he d l f f e o m o r p h i s m t y p e of the a c t i o n is c o n s t a n t .

§8 . D e f o r m a t i o n s of n o n - d e g e n e r a t e h o l o m o r p h i c m a p s

A s s u m e we a r e i n t h e f o l l o w i n g s i t u a t i o n : we a r e g i v e n a f a m i l y of d e f o r m a -

t i o n s of M, I~: Z -~ ]:3, a n d l e t r s a s s u m e t h a t , W b e i n g a f i x e d c o m p l e x m a n i f o l d ,

one is g i v e n a h o l o m o r p h i c m a p F: Z -~ W ×]B, s u c h t h a t R = PZ 0 F , PZ: W -~ }5

b e i n g t h e p r o j e c t i o n on the s e c o n d f a c t o r of the p r o d u c t . T h i s g e n e r a l s i t u a t i o n h a s

been considered by Horikawa ([Hor 0], [Hor I], [Hor Z]), here we shall limit our-

selves to the case when

= o [ : Xtc-~W is generically with injective differential (ft)~ (8. I) ft Pl F Xt

( h e r e P l : W × B -~ W is t he f i r s t p r o j e c t i o n ) .

( 8 . 1 ) i s e q u i v a l e n t to s a y i n g t h a t V t (ft)~ ®Xt -~ (i t) ®W i s a n i n j e c t i v e h o m o -

of s h e a v e s , w h e r e (i t) ~' d e n o t e s t h e a n a l y t i c p u l l - b a c k of a c o h e r e n t m o r p h i s m

sheaf. In particular, for t = 0, we have an exact sequence

(fo)~ .-~ ( 8 . 2 ) 0 -~ ®X ~ (fo) ®W -~ Nf -~ 0

o o

D e f i n i t i o n 8 . 3 . The c o k e r n e l Nfo of t he h o m o m o r p h l s m (fo),.~ in (8. 2) i s c a l l e d

t h e n o r m a l s h e a f of t h e h o l o m o r p h i c m a p , a n d H°(Nfo) i s c a l l e d the c h a r a c t e r i s t i c

s y s t e m of t he m a p .

P r o p o s i t i o n 8 . 4 . T h e r e i s a l i n e a r m a p OF: T B , o -~ H° (Nfo ) s u c h t h a t , if P is

t h e t ~ o d a i r a - S p e n c e r m a p of t he g i v e n d e f o r m a t i o n , a n d @ is t he c o b o u n d a r y m a p

8: H° (Nfo ) -~ H I ( ® x o ) of t he l ong e x a c t c o h o m o l o g y s e q u e n c e of (8. 2), t h e n one h a s

a f a c t o r l z a t i o n O = O o PF " Such a O F i s c a l l e d t he c h a r a c t e r i s t i c m a p of t he

family.

Proof. Let's take a finite cover (V¢~) of Z such that Vc t _-m Uc× B as usual, so ¢t

c X , we can write that, locally on V(~, if z are local coordinates in U¢¢ o

(8.5) F ( z ~ t ) = ( f ( z ~ t ) , t )

Let ~ be a tangent vector in TB,o which can be extended as a vector field on B

25

(we a l w a y s w o r k wi th (B, o) a s a g e r m of c o m p l e x space ) , and l e t , ~ be the l i f t of

to Vc~ d e t e r m i n e d by our cho ice of c o o r d i n a t e s (z c~, t) on VC~. We know that

t , P(~) = ( ~ a - #~) X ° and tha t in fac t ( s ince one can change c o o r d i n a t e s ) , z9 a is we l l

de f i ned on ly up to add ing a v e r t i c a l v e c t o r f i e ld . The d i f f e r e n t i a l F$ s e n d s @C~ in

a pair (q~C~' ~) where q~c£ is a sectionof f (@W). #

Restricting ~c~ to Uct X {0 } , I get a global section ~ of Nfo = fo(®W)/®Xo

and by definition, since (fo).~(~gC~IUcc X [0]) = d20 c , we get

a ( ~ ) = (0-~) x = P ( ~ ) •

o

So it s u f f i c e s to se t PF(g) = 4 , and PF is w e l l - d e f i n e d and l i n e a r . Q.E.D.

The situation considered up to here embodies the classical theory of deforma-

tions of plane curves with nodes and cusps, and of surfaces with ordinary singulari-

ties, therefore we shall now give a definition which is consistent with the classical

one in the second case, but is not a generalization of definition 5.1 (also we shall

denote by CO the map ~0: M -+ W corresponding to f via the isomorphism M m X). o

Definition 8.6. The characteristic system H°(% ) of the map ~0 : M -~ W is com-

plete if there exists a smooth deformation of the holomorphic map (i. e. B is

smooth) such that PF: TB, o -~ H°(N~ ) is surjective.

Remark 8.7. H°(NCO) is the exact analogue of HI(® M) in the case of deformations

of a manifold. In general, a sufficient condition in order that the Kuranishi family

be smooth is the sharp assumption H2(®M) = 0 : in a similar fashion Horikawa

proves the following generalization of a previous theorem of Kodaira:

Theorem 8.8. The characteristic system of a map is complete if HI(NCO) = 0 .

In the next paragraph we shall discuss some particular example in the special

case when CO is an embedding: before doing so, we just make the following observa-

tion (in view of the long exact cohomology sequence associated to (8. Z)).

(8.9) a necessary condition in order to deform CO ona complete family

I ~ HI(NCO ) of deformations of M is that H (¢O (@W)) -> be

injective.

26

§9. Examples of embedded deformations and obstructed moduli

Assume now that M is a subvariety of W, that dim W = r, dim M = n, and

that M is the locus of zeros of a section of a rank (r-n) vector bundle Vo We

shall write VIM for @W (V) ® @M ' and it is an elementary computation to see l

that, if the NMI w denotes the normal sheaf for the embedding of M into W, then

(9.1) NMI w : v i m ,

h e n c e we have an exact s e q u e n c e

0 -+ 9 M -+ ®W @ @M -~ VIM -~ 0

and clearly the characteristic system is complete if there is a surjection

H ° ( W , V ) - ~ H ° ( M , V i M ) .

Example 9. 2.

r - n

W = ~ r , V = ® @ (m.) i=l ~r

(m. -> 2) . 1

The ideal ~ admits a Igoszul resolution M

(9.3) 0 -~ Ar-n(v V) -~ Ar_n_ I V V A2(W) V • -~ 0 ( ) -~ .. -~ -~ V -~ 3 M

dual to Akv

A(f,} fn r A k+l V, the f.'s being the s e c t i o n s

1

of @~r (mi) s . t . M = {f l . . . . . f n - r = 03

we infer that H°(~ r, V) goes onto

F r o m (9 .3 ) , by i nduc t ion , f o l l o w s tha t

i (9.4) H (3M(k)) = 0 ¥ k E • and for i <- n

In particular, HI(3 M • V) = 0, hence, by the sequence

0 -* 3 M V -~ V -~ V I M - * 0 ,

H° (M, VIM) . A l s o , f r o m the s e q u e n c e

0 -~ 3M(k) -~ @u?r(k) -~ @ivi(k) -+ 0

f o l l o w s that

27

i Hl(v I (9 .5) H ( @ M ( k ) ) = 0 V k E ~ if i ~ n - 1 ( e . g . M ) = 0 )

It w i l l be t r u e tha t the f a m i l y g i v e n by H°(V) is c o m p l e t e if and on ly if , in v i e w of

the long e x a c t c o h o m o l o g y s e q u e n c e a s s o c i a t e d to (9. 1), HI (® r ® f9 M) = 0. Now,

in v i e w of the E u l e r s e q u e n c e (6 .3) , and of (9 .5 ) , we h a v e an e x a c t c o h o m o l o g y

s e q u e n c e g i v i n g

(9 .6 ) Hl (®lp r ® O M) = 0 if n >- 3, o r i f n = 2 and

H2(@ M) -~ H2(@M)r+I i s i n j e c t i v e .

i for @((A I 1,0)V But in this last case (we use the standard notation f~Ni T M )), by

Serre duality an equivalent condition is that the following map be surjective

2 1) ) r+ l 2 H°(flM(- -~ H°(g2M ) .

2 ~ @M(2 m i r - 1), and s i n c e H°(@p(k) ) -~ H°(@M(k)) is But , by a d j u n c t i o n , 0 M

onto V k, we get that

r-2 2

--~ @M ( i . e . Z m.1 = r + l (9 .7 ) H I ( ® r ® @M) = 0 u n l e s s GM 1

an e q u a t i o n which has on ly the f o l l o w i n g s o l u t i o n s fo r the m . ' s : (4), (3, 2), (2, 2, 2)). 1

c a s e HI(@ M) = H°(fl~)--~,~ = 0 (M is a Kffhler m a n i f o l d v ia the F u b i n i - In this last

2 m @S H2(®M) is dual by Serre duality to Study m e t r i c ) , but , s i n c e f2 M

HO((f11) ® f/Z) = HO(f)l) = 0 .

The c o n c l u s i o n is the f o l l o w i n g w e l l known f a c t (Cfo [Se r ] ) .

(9.8) If M is a ( smoo th ) c o m p l e t e i n t e r s e c t i o n in &or of d i m e n s i o n n_> 2,

the c h a r a c t e r i s t i c s y s t e m is c o m p l e t e , and the e m b e d d e d d e f o r m a -

2 ~--@M: M t i o n s g ive a c o m p l e t e d e f o r m a t i o n , e x c e p t i f n = 2 and f?M

is c a l l e d t h e n a 143 s u r f a c e , and e m b e d d e d d e f o r m a t i o n s g i v e a 19-

d i m e n s i o n a l s u b v a r i e t y of the N u r a n i s h i f a m i l y , w h i c h is s m o o t h of

d i m e n s i o n 20.

So, fo r i n s t a n c e , e v e r y s m a l l d e f o r m a t i o n of a s m o o t h s u r f a c e S of d e g r e e m in

IP 3 is s t i l l a s u r f a c e in ~3 : bu t wha t h a p p e n s in the l a r g e , a c c o r d i n g to the f o l l o w -

ing d e f i n i t i o n ?

28

Definition 9.9. Two manifolds M, M' are said to be a deformation of each other

(in the large) if they lie in the same class for the equivalence relation generated by:

M Def M' ~ there exists a deformation: K : 75 * B of M with irreducible B

and such that 3 b E B with X b isomorphic to M' If M Def M', we shall also

say that M' is a direct deformation of M . The upshot ([Hor 3]) is that already

for degree n=5 the surfaces which are deformations of quintic surfaces need not be

surfaces in ~p3 any more !

The rough idea is as follows : Horikawa considers all the smooth surfaces

such that pg = h°(f~2) = 4, and such that K 2 = 5 (K, here and in.the following, is a

divisor of a (regular here, rational in other cases) section of f~S): these numerical

conditions, as we shall see, are invariant under any deformation, and indeed any

complex structure on a surface orientedly homeomorphic to a smooth quintic surface

in ;p3 must satisfy these conditions.

Studying the behaviour of the rational map ~P : S -> [p3 associated to the sec-

Z Horikawa shows that either tions of flS '

I) <p g i v e s a b i r a t i o n a l m o r p h i s m o n t o a 5 - i c o r

I /a ) ¢P g i v e s a r a t i o n a l m a p 2 : 1 o n t o a s m o o t h q u a d r i c o r

lib) <P is 2:1 to a quadric cone.

Note that for a smooth 5-ic ~2S -~ (9S(I)' hence ¢p is just inclusion in IP 3 and

one is in case I. Surfaces of type I belong to one family of deformations, and their

Kuranishi family is smooth of dimension 40: the same holds for surfaces of type IIa).

For surfaces of type I~0), instead, h1(®S) = 41, h2(®S ) = i therefore we know that the

Kuranishi family gives a hypersurface in (~41 : I-lorikawa computes

[ , ] HI(®s ) × HI(®S )* HZ(®s ) ,

finding that, in suitable coordinates, the associated quadratic polynomial is z I z 2 .

Now, by the Morse lemma, there do exist new coordinates on HI(@s ) s.t. the equa-

tion g of B is of the form g(t) = t It Z +~(t 3 .... ,t41), where @(t) = o(t2). Since

H°(®S) = 0, the Kuranishi family is universal and surfaces of type I]b) form, as it

is easy to show, a 39-dimensional variety which is contained in the singular locus

of B by Horikawa's computation of [ , ]. Now the singular locus of B is given

by t I = t 2 = 8~/8t 3 ..... 8~/8t41 = 0, hence it has dimension 39 iff ~ -- 0, Thus

g = glg 2, with gi(z) = z i+ o(z). The conclusion is now easy: when gl = 0, g2 ~ 0

we have a surface of type I (a 5-ic), when g2 = 0, gl ~ 0 we have a surface of

type IIa, when gl = g2 = 0, we have a surface of type IIb.

29

We e n d t h i s s e c t i o n s h o w i n g a n i c e e x a m p l e due to M u m f o r d ( [ M u 1] , cf .

a l s o a p p e n d i x to C h a p t e r V of [ Z a ] ) of v a r i e t i e s f o r w h i c h the K u r a n i s h i f a m i l y is

e v e r y w h e r e n o n r e d u c e d . We n o t i c e t h a t the t e r m i n o l o g y m o s t f r e q u e n t l y u s e d

a d o p t s the f o l l o w i n g

Definition 9. i0. A manifold M is said to have obstructed moduli if dim B <hi(@)

(if and only if B is not smooth), i. eo if "not all infinitesimal deformations are

integ table. "

It frequently occurs that B may be singular at 0, but the phenomenon

pointed out by Mumford is not so common (at least for the time being).

Example 9. II. ([Mu I])o Let F be a smooth cubic surface in [p3 , E a straight

line contained in F (hence } Z : -I, KE : -i), and let H be a hyperplane section of

F° The linear system 14H + ZE I has no base points (this is clear outside E, on

the other hand by the exact sequence 0 -> @F(4H) -~ @F(4H+E) -~ @E(3) -~ 0 since

1 HI(@F(4H + E)) = 0, H (@F(4H)) : 0, we get that 14H 4 E l has no base points and

then we conclude by the exact sequence 0 -> @F(4H+ E) -~ @F(4H + ZE) -~ @E(Z) -> 0,

where @E(i) is the sheaf of degree i on E --~ iDI), so that we can pick a smooth

curve C inside 14H42EI. Since the canonical sheaf of C is @c(3H +ZE), we

easily find that

(9. IZ) C c F is a s m o o t h c u r v e of g e n u s g : 24 a n d d e g r e e 14.

S ince the n o r m a l s h e a f of C in F , NC} F is @c(4H + ZE) , w h i c h is n o n s p e c i a l ,

we g e t a n e x a c t s e q u e n c e of n o r m a l s h e a v e s

HO(Nc, l e3) o (9. 13) 0 -~ iF ) -~ H°(N -~ H (@c(3H)) -~ 0 C

~3) -- ~ HI((gc(3H)) ' and also HI(Nc] a dual vector space to

dimension 1 by virtue of the exact sequence

H°(@c(ZE) ) , w h i c h h a s

0 ~ H°(@F(ZE)) ~ H°((~c(ZE)) -~ HI(@F(-4H)) = 0 °

We see t h a t the h y p o t h e s i s in t h e o r e m 8 . 8 i s no t v e r i f i e d , a n d in f a c t the c h a r a c t e r -

i s t i c s y s t e m is no t c o m p l e t e , a s we s h a l l s e e . (9. 13) g i v e s d i m ~ H ° ( N C l io3 ) = 57,

m o r e o v e r F m o v e s in a 1 9 - d i m e n s i o n a l l i n e a r s y s t e m i n IP 3, C v a r i e s lin a 37-

d i m e n s i o n a l l i n e a r s y s t e m on F , h e n c e C b e l o n g s to a 5 6 - d i m e n s i o n a l f a m i l y .

M u m f o r d s h o w s t h a t C c a n n o t b e l o n g to a n a l g e b r a i c f a m i l y of d i m e n s i o n 57 b y t h e

f o l l o w i n g a r g u m e n t s :

30

I) if C' is a smooth curve of genus Z4, degree 14, @C,(4H) is non special

and has 33 independent sections, so that C' belongs to Z independent quartic sur-

faces G, G'. Clearly, C' is not a plane curve, and it is easy to check that C is

not contained in any quadric surface.

Z) Thus, e ithe r

a) C' is not contained in any cubic surface or

b) C' belongs to a (unique) smooth cubic surface or

b I) C' belongs to a singular cubic surface.

Assume now that C belongs to an irreducible family of dimension • 57: 3)

since condition b') is a closed condition on the base of the family, there would exist

a family of curves C' of type a), with dimension • 57. But in case a), G • G' =

CS~ F, where F is a conic. Thus the complete intersection of G, G', being

Cohen-Macauley, is reduced and has at most triple points as singularities, so that

G and G' have no singular points in common. Since G and G' intersect trans-

versally at the points of C n- F, we can assume G to be non-singular around C' .

4) By Noether's theorem, not all surfaces of degree zi contain a conic, hence

G belongs to a family of dimension at most 33 (in fact, much less, see [G-HI),

moreover it is easily verified that the characteristic system of C l in G has dimen-

sion 24, so that the dimension of such pairs (C' c G) is at most 57, and since C'

belongs to a 1-dimensional system of quartic surfaces, we are done.

Example 9. 14 ([14o I], [Mu i] ). If M is the blow-up of ~ °3 with centre a curve C

as 9. ii, then the base B of the IKuranishi family of deformations is non reduced.

Before even setting up the notations, let's give a useful

Definition 9. 15. Let Y be a subvariety of a smooth variety X. We define ®X

(-log Y) to be the sheaf of tangent vectors on :K which are tangent to Y (i.e. ,

E @X (-log Y) if and only if, ~gy being the ideal sheaf of Y , Y g E ~Jy~

~(g) ~ Jy).

Remark 9. 16. Clearly, 3y® X c ®x(-log y) and, by the definition, ®X/®x(-log Y) l

' of Y in X. (NyI X is the usual normal is the equisingular normal sheaf Ny] X

sheaf when Y is smooth, cf. §II.) We have thus the exact sequences

®X ( - l o g ' 0 (9. 17) 0 -~ Y) -~ ®X ~ Nyix ->

Moreover the tangent sheaf of Y , ®y is by definition the quotient

31

®x(-log Y ) / 3 y ® x "

Let's now set up the notation: w e have 11 : M * ~3 the blow-up map, E is

the exceptional divisor II" I(C).

We have therefore two exact sequences

0 -~ ®Bo 3(- log C) * (®Eo3) -~ NC/~O 3 -+ 0

0 -* ®M('l°g E) -* ®M -* NE[M -~ 0

First of all, by the exact sequence

0 -~ ~C 0[03 -~ ®[p3 (-l°g C) -~ ®C -~ 0

since C has genus bigger than 2, H°(®C ) = 0, thus also H°(®ip3(-log C)) = 0 o j

since H ( C ®ip3 ) = 0 (this is the Lie algebra of the group of projectivities leaving

C fixed, which is the trivial group since C contains 5 independent points).

Let's look at the exceptional divisor E : E is the projectivized normal

bundle of C in IP 3 , E = IP(NcIIp3), i.e. points in IE are lines in the normal i

bundle of C° In particular, if @E(1) is the dual of the tautological invertible sheaf ~ v

@E(,I) c 11 (NcI[m3) , then ri~@E(l ) = NCI[O 3 (v denoting the dual sheaf).

It is easy to verify that

(9.18)

hence we have

(9.19)

NE] M -~ @E(.1) ,

H°(NE]M ) = HI(iNEIM)= 0 .

Proof of (9. 19). Since a sheaf of degree (-i) on has 0 cohomology in all

= f~l degree , I I NE1M [I:~ NE[ M = 0. Q . E . D .

• i Corollary 9.20. HI(®M) --~ H (®M(-log E)) for i = 0, 1,2o

Proposition 9.ZI. 11 , eM(-log E) C), ~111 . ® E) = 0, ~ = ®&o3(-log ,~ M(-l°g

Hi(®M('l°g E)) --~ Hi(®o 3(-log E)) for i = 0, 1,2

o and H (®M(-log E)) = 0 m

hence

The proof of proposition 9. Z 1 follows immediately from the following.

32

Lemma 9. gg. Let [[: X-+ C Z be the blow up of the origin O, ~ the maximal

ideal of the point O, E the exceptional divisor [I" i(0). Then

I I , : ,®x(_log E) = D ®C2 , II,~@x(-log E) = 0 .

Proof. X is covered by two affine pieces A, A' wDah coordinates (u,v), resp.

(u',v') and [I is given by x = u, y = uv (resp. x = u'v' , y = u'). Thus

Since

la o a__ {a ,a au - ax 4 - v ay = ~yy + v a~

l a ~ _ _ u,a = u ay V av'

Ig is defined by the equation u = 0 (u ' = 0), ® (-log N.) is generated by

a a a a O U ~ u = x ~ x + Y ~ y a n d b y ~ = x ~ y on A, a n d b y

, ~ a O O a A ' U ~ = y ~y + x ~xx and --av ' : y ~x on

Let ~ be a section of ®x(-log ]g) defined in a neighborhood of E: then we can

express ~ as

a = a ( u , v ) u ~u + b ( u , v ) ~v and a l s o

~ = C~(u',v')u'# + ~(u',v')~ , w h e r e

a ( u , v ) = E a . . uiv j i , j ~ 0 U

and similarly for b, ct, ~ . We must have

o~7- + (a y • b x ) oyZ-- = ( ~ x + ~y) ~ + ( w ) ~__ • o y

H e n c e , e x p r e s s i n g ~ as a r a t i o n a l s e c t i o n of ®(g Z , we s e e tha t the c o e f f i c i e n t of

8 / 8 x can be any f u n c t i o n f of the type

x • E a..x i-j yJ , i,j >0 U

such that it can also be expressed in the form

X " E h i-h

~hl x • y h,i >0

~m x h ~ -h + Y 8h,~ Y

h , 0

33

It is immediate to see that the coefficient of 8/8x is a power series in x, y, van-

ishing at the origin, and, by symmetry, the same holds for the coefficient g of

8/0y. It is also easy to verify that two such f,g E ~@ Z can be chosen arbi- 1 C ,0

trarily. Instead, to show that R II :.~(®x(-log E)) = 0, it suffices to show that any

x(-log A + ' A + section A of ® E) on A N A S can be written as a sum +A , where is

regular on A, At is regular on At. Now, A can be written as

whe re

can be written as

+ b(u,v) V4 '

I i vj a(u,v) : a. u i ~ 0 U j£~

a4 (u,v) + ~ i i a(u,v) : +a'(u,v) a = a.. u v i,j >0 zJ

and similarly b(u,v) = b+(u,v) 4b'(u,v). Since

A + + O b + 3 :a u~+ 0v

is regular on A, it suffices to verify (we omit this) that A' is regular on A' .

Q.E.D. From 9.20 and 9. ZI we infer that

H°(®M ) = 0, HZ(®M ) --~ HI(NcI~3) , HI(®M )-~ H°(l'qcllp3)/H°(®lp3 ) .

These isomorphisms are natural, in fact one can verify that for each deformation

of the embedding C ~IP 3 the Kodaira-Spencer map for the family of blown up

3-folds is the composition of the characteristic map of the deformation with the sur-

o l o jection of H (INC]Ip3) -~ H (®M) (the kernel H (®E~3) is due to the fact thatblow-

ing up projectively equivalent curves one obtains isomorphic 3-folds). Now,

Kodaira ([Ko I], thm. 6) proves that every small deformation of M is the blow-up

of IP 3 with center a curve which is a deformation of C in 3, thereby showing

that the Kuranishi family of Ivl has dimension equal to 56-15 = 41, whereas, by

whatwe saw, hl(@ivi) = 42 for each blow-up M of a curve C as in 9. Ii. Thus

the Kuranishi family B of M is singular at each point (hl(@Mt) being constant

for t 6 B , B is the Kuranishi family for each Mt).

34

§I0. Further variations and further results

We have seen in {9 a 3-dimensional variety such that its iKuranishi family is

universal at each point, but its base B is everywhere non reduced. We remarked

in §5 that the base B of the Kuranishi family of curves is smooth: for surfaces

Kas (if<as]) found, using Kodaira's theory of elliptic surfaces, an example of a

family of elliptic surfaces such that the generic dimension of 51(®$% ) would be

strictly bigger than dim B. The family is constructed by deforming a certain

class of algebraic surfaces. %%re suspect that this should not happen for surfaces of and }( an~,ple

general type with HI(S,~) = 0V(cf. lecture seven); it is anyhow clarified by Burns

and Wahl ([B-V/]) how the fact that K S is not ample, in particular the existence

of many curves ~ such that ~E(KS ) ~- ~E ' forces the dimension of HI(®) to be

bigger than dim B : in particular, using classical results of Segre on the existence

of surfaces • in ~3 with many nodes (also called conical double points, i.e. with 2 2 2

local equation x + y + z = 0), they show that the blow-up of ~ at the nodes is a

surface S with obstructed deformations (the rough idea being that nodes contribute

by 1 to hl(®), but all the small deformations are still surfaces in [p3). We al-

ready noted in the introduction that Zappa ([ Zp]) was the first to show that the

characteristic system of a submanifold does not need to be complete. His example

is as follows (we follow, though, the description of []Y[u 311:

Example i0. i. Let E be an elliptic curve and let V be the rank Z bundle which

occurs as a non trivial extension

( 1 0 . 2 ) 0 -> (9 E -~ V ÷ ~E ~ 0

* H I ( 0 E ( in f a c t , t h e s e e x t e n s i o n s a r e c l a s s i f i e d b y G o r b i t s in ) ~ C , so t h e r e is

" o n l y " one n o n t r i v i a l e x t e n s i o n ) . T he s u b b u n d l e •E d e f i n e s a s e c t i o n C of t h e

° l b u n d l e S = ~ ( V ) o v e r E , a n d , s i n c e V / @ E ~ - ~ E , N C I S ~ @C" N e v e r t h e -

t h e r e is no e m b e d d e d d e f o r m a t i o n of C in S, s i n c e H I ( @ s ) m H I ( @ E ) , and , l e s s ,

i f C t is a l g e b r a i c a l l y e q u i v a l e n t to G, t h e n t h e r e is a d i v i s o r L of d e g r e e 0 on

E such that (II: S -~ E being the bundle map) C l -C +~'*(L). But then

[ I , ( @ s ( C t ) ) = V ® (gE(L). a n d , t e n s o r i n g 1 0 . 2 w i t h (gE(L). we i n f e r t h a t t h e r e a r e

no s e c t i o n s if L ~ 0, w h e r e a s , f o r L =- 0, t h e c o n d i t i o n t h a t t h e e x t e n s i o n s p l i t s

e n s u r e s t h a t 1 0 . 2 i s n o t e x a c t on g l o b a l s e c t i o n s (o r g e o m e t r i c a l l y , if

h O ( ~ s ( C ) ) ~ I ~2, S~E× !) . Q.E.D.

35

As far as deformation theory is concerned, the Kodaira-S~pencer-Kuranishi

results were extended first in the direction of the deformations of isolated singular-

ities (cf. [Po], [Gr I]), and then the result of Kuranishi was extended to the case

of compact complex spaces ([Gr 3], [Dou 3], [Pa]). On the other hand, Grothen-

dieck ([Gro l], [Gro Z]) contributed significantly to extension of the deformation

theory, especially through the construction of the Hilbert schemes, parametrizing

projective subschemes with fixed Hilbert polynomials (cf. §19 for a vague idea):

his results were extended to the case of (compact subspaces of) complex spaces in

Douady's thesis ([Dou 4])° Since the variations in the theme of deformations can

be arbitrary, Schlessinger ([Sch]) approached the problem abstractly developing

a general theory giving necessary and sufficient conditions for finding "power

series solutions" , i.e. finding a formal versal deformation space for a deforma-

tion functor: this theory is usually coupled with a deep theorem of Artin ([Ar]),

giving criteria of convergence for the power series solutions. ~Ve don't try to

sketch any detail, nor to mention further very interesting work, but we defer the

reader to the very interesting article ([Pa]) of Palamodov already quoted in the

introduction (and plead guilty for ignoring the post '76 period). We simply remark

the importance of Palamodov's theorem 5.6 giving an algebraic description of the

higher order terms in the K(~ranishi equations.

As far as Iknow, this result has not yet been applied in concrete geometric

cases, but its validity should be tested in some example.

36

LECTURE FOUR: THE CLASSICAL CASE

§11. D e f o r m a t i o n s of a m a p and e q u i s i n g u t a r d e f o r m a t i o n s of the i m a g e ( i n f i n i t e s i r n a l the® ry )

We l e t , a s in §9, [0: X-~ W be a n o n - d e g e n e r a t e h o l o m o r p h i c m a p , a n d we

s e t E = O(X). S i n c e W is a s m o o t h v a r i e t y , we h a v e in g e n e r a l , fo r e v e r y s u b -

v a r i e t y E, the e x a c t s e q u e n c e s

t 0 -~ ® w ( - l o g Z) -~ ® -~ NZ I W -+ 0

(11. 1) W

0 ~ ® W ( - 2 ) -~ ®W(-Iog Z) -* ®E -~ 0 o

On the o t h e r h a n d , b y d u a l i z i n g ( i . e . , t a k i n g

V w h e r e NE I W

Horn ( • , @Z)) 0 Z

v ~/i ® 0 -~ f~l 0 0 -~ N E [ W -~ W E E -~

the exact sequence

is the c o n o r m a l s h e a f of g in W, we g e t t he long e x a c t s e q u e n c e

V -'[¢

o - ~ e E ~ ®w ® °z ~ ( N z J w ) ~ NZlW 1

-~ E x t l ( f2Z, @Z) -~ 0

w h i c h s p l i t s in to the s h o r t e x a c t s e q u e n c e s

(ll.Z) 0 -~ ®E -~ ®W ® C3E -~ NE [W -~ 0

0 - NZ'[W " NZ[W " Ex t l (nz l ' OZ ) -~ 0

Example ll. 3. Assume E is a hypersurface in W, locally defined by the equa-

l tion f(x l ..... Xn) = 0. Then N E I W ~ OE(Z), and N x is the subsheaf defined by

(Of/0x I' ' ' then the ideal sheaf ... 8f/SXn )" Thus if g is a section of N x I W ,

ft = f(x) + tg(x) = 0 gives an infinitesimal deformation of E which is "equi-

singular," i.e. , modulo (t2), the locus of zero has not changed. In fact, if

g(x) = E af . ~ • ui(x) , l. 1

then, setting u(x) = (Ul(X) ..... Un(X)), we have that f t(x) -= f(x + tu(x)) (rood t Z) by

T a y l o r e x p a n s i o n .

Definition ll. 4. The morphism %0 is said to be stable if the direct image sheaf

~0~(N%0 ) is isomorphic to the equisingular sheaf N' , .,. ZlW

i Remark 11.5. Bylooking atthe stalks of INEI W and of ¢@;:.~(Nqo) at p , a smooth

point of E, such that ~0 has differential of maximal rank at the points in ~0-1(p),

37

we see immediately that if q) is stable, then ~ is birational onto its image (other-

wise one should equip the locus E with a scheme structure with nilpotent elements).

We shall assume from now on ~p to be birational onto its image.

P r o p o s i t i o n 11 .6 . A s s u m e d i m X = 1, and t ha t p is a s i n g u l a r po in t of ~2: t hen

if ~0 is s t a b l e , then p is an o r d i n a r y doub le p o i n t (nod__~e) and d i m W = Z.

Proof. In fact, the rank of N~ [W at p (rankp 3 = dimc~/~p~, Dp being the !

maximal ideal of p) is n=dimW, since 8/8x I, 8/8x 2 ..... 8/8x n generate NEI w local-

ly, while vector fields in ®E vanish at p (cf. [Ro] , thin. 3. Z); whereas the rank of

~(N~0) at p is just the sum (N <p)q

d i m ~ <p(q) = p #

~o g~p (N<p)q

If q0(q) = p, then we can take l o c a l h o l o m o r p h i c c o o r d i n a t e s t a t q ,

(Xl ..... Xn) at p , such that

a l al+a 2 al+ a2 +...+a ) r

~p(t) = t , t + . . . . . . . . t + . . . . . . 0 , 0

where a. > 0, and r is the smallest dimension of a local smooth subvariety contain- i

ing the branch of g corresponding to q (clearly, al=l<==>r =I, and r>al). It is easy

then to see that dim(N~) q

ana I - i ~0 ~p(IN%o)q

Hence if this dimension has to be less than n, we get a/ready a I = I; moreover,

since the sum over all these points q has to be less than n, we infer that n = Z and

there are exactly two smooth branches. Either the two branches are transversal,

and we have a node (:<ix Z = 0 in suitable coordinates on %V), or we have a double 2 Zk+Z

point of type (we set x =x I, y =xz) y =x (km I). In this case

~ , ( N )

~2 ~o,(N ) ' P

though, has dimension 4 whereas

N Z , p (y, x z k ÷ l )

Z N (Y, x Z k + l ) ( y Z , Zk+Z) xy, x z) + (yZ.x p E,p

g 2k+l Zk+l has dimension 5 (since y, xy, y , x , x • y are a C-basis). Q.E.D.

38

Example 11.7. The morphism t ~+ (tZ,t3), giving an ordinary cusp, is not stable

since in fact the deformation t ~ (a 0 + t 2, b 0 } blt + t 3) gives a node: in fact

8Xl/0t : Zt, 8xz/St : b I + t Z , and t : ± ~ are two points of X mapping to

the s a m e point.

R e m a r k 1 1 . 8 . We r e f e r to [ M a t h 1, Z] f o r a t h o r o u g h and g e n e r a l d i s c u s s i o n a b o u t

s t a b l e m a p g e r m s : h e r e we s h a l l l i m i t o u r s e l v e s in t h e s e q u e l to d i s c u s s o r d i n a r y

singularities when dim X : Z. Before dealing with this special case, let's see what

is true in general.

t Theorem 11.9. There is a natural injective homomorphism of N E IW to <0. N ,

to be f i n i t e and b i r a t i o n a l on to Z . if you assume ~9

Proof. We have the two following exact sequences, with the homomorphism

induced by p u l l - b a c k cO : ( 9 -~ ~ , @X

(11. I0) I

0 -~ ®E -> ®W ® @E -~ NE]W -> 0

l, -~ e l ~ . , ® X , , ` = 0

andwehave to verify that ~(® ) c ~* ®X" I.e., this is whatwe need to verify:

if d E is the ideal sheaf of Z, and (x 1 ..... Xn) are coordinates in W, whenever

al(x) ..... an(X) are functions such that ~' f E d E ,

n

E ~ E 8f

• = I i i ai(x) 8x.

then there do exist, for each point y s.t. ¢P(y) = x, functions $1(y ) ..... ~m(y )

(m = dim X, y = (Yl ..... Ym ) are coordinates on X) such that

m

Since X is s~ooth, by Hartogls theorem it will suffice to show the existence of

such functions outside a subvariety F of codirnension at least Z in X.

We first remark that, since we are assuming ~p to be birational onto 52,

the subvariety Z c X where <p is not of maximal rank has image tp(Z) c Sing(E) n n

(if ~: IE 0 -> (E 0 has local degree i, then it is a local biholomorphism). If

39

x 6 Z - S i n g ( Z ) , t h e r e i s n o t h i n g to p r o v e , o t h e r w i s e t h e r e e x i s t s a c o d i m e n s i o n 2

A of Z with subvariety

i)

ii)

A c Sing(Z) , 1 m-i

if x £ Sing(E) -A, Z is locally biholomorphic to Z X C where 1 n-m+l

is a curve in ~]

iii) ~0-1(A) = F has codimension at least 2 in X (this follows from the

assumption that ~0 be finite,

iv) if x E Sing(E) - A, y E ~-1(x), then there are coordinates (Yl ..... Ym )

around y such that, using the local biholomorphism 52 -~ Z 1 X C m" 1 •

q) (Y) = (~°I(Y)' YZ ..... Ym )"

By our previous remark and iv) above, it suffices to prove our result in the case

when dim X = dim Z = I. In this case, we denote by t a local coordinate at a

point y of X, according to tradition, and we may assume that, g 1 <i < j < n,

projection on the (i, j) coordinates maps Z birationally to a plane curve of equation

Fij(xi,xj) = 0. Since Fij 6 ~Z ' w e have

8F OF ai(x(t)) ~ (xi(t),xj(t)) + aj (x(t)) ~ (xi(t), xj(t)) ~- 0

A s s u m e w e s h o w t h a t t h e r e e x i s t s V i , j , a m e r o m o r p h i c f u n c t i o n v ( t ) w i t h

) d~.(t) a . ( x ( t ) ) = v ( t ) d x i ( t - , a . ( x ( t ) ) = v ( t ) J t dt j dt

then all the (2x2) minors of the matrix

i (t) ~n(t) dt """ dt

a 1 (x(t) . . . a n (x {t))

vanish, and we can conclude that there is a v(t) with a.(x(t)) = v(t) (dxi(t)/dt) for 1

each i , p r o v i d e d t h e f o l l o w i n g h o l d s t r u e .

Lemma i. IZ. Let ~2 be a germ of plane curve singularity, with equation f(x,y) = 0

and let X = ~l(t), y = q0z(t ) be a parametrization of a branch of ~. Then, if al(t),

a2(t ) are functions such that

of (t) 8f al(t) ~ (Vl(t), q)Z(t)) t a 2 7y (cPl(t)' CPz(t)) ~ 0 ,

there does exist a meromorphic function v(t) with

40

d cp. (t) 1

a.(t) : v(t) - - (i = I,Z) i dt "

Proof.

Clearly

Write f = flf2 where fl = 0 is the local equation of the given branch.

Of Of 1 Ox (~i (t)' ~Z (t)) :-~-~ (~I (t)' ~Z (t)) " f2(~i (t)' ~2 (t))

and analogously for 0flOy: since fz(q~l(t), ~02(t ) ~ 0, we can indeed assume

have only one branch. Without loss of generality we may assume

¢Pl(t) = t m , OZ(t ) = g ( t ) = t m + c + . - "

Z to

(m is the multiplicity, c the local class). As classical, we use a base change CZ cZ m

¢p: -~ sending (t,y) to (x = t ,y): then o-l(z) consists of m smooth

branches, of equation y- g(te i) = O, with e = exp(ZTr~-~-T/m), i = 1 ..... m (the

first of these branches coincides with the given parametrization). Clearly the pull-

backs ~0~v(Of/Oy) and ~3"~(Of/Ox) coincide, respectively, with

Since

by assumption

af ( t ,y ) and ~ a f ( t , y ) Oy m - 1 Ot

rnt

m

f ( t , y ) = ~ (y - g( tc~) ) , i = I

O f ( t , y ) m - 1 8 f ( t , y ) a l ( t ) 8t + a z ( t ) m t 8y

v a n i s h e s i d e n t i c a l l y a f t e r p l u g g i n g in y = g ( t ) . We g e t

m

E Tr i = l j / i

• i 8g m t m - 1 (y - g( teJ) ) • ~ (to i) + az(t) m

i = 1 j # i

a n d p l u g g i n g i n y = g ( t ) , w e o b t a i n

N o w

ag (t) 1 -al( t ) ~ + a z ( t ) m t m"

dxi(t) a i ( x ( t ) ) = v ( t ) dt

- 0

Q. E . D. f o r t h e 1 e m m a .

let m be the multiplicity of the branch (i.e. m = rain ord t xi(t)) ) and assume

xl(t ) = t m : then ord t ai(x(t)) > m, hence ordtv(t ) ~ 0 and g is holomorphic.

It remains to be proved that the given homomorphism of N E IW into ¢p~ Ncp is

41

injective. Inview of (ll. lO) we have to show that if a section ~ of ®Vf ® @Z lies

in the image of %O, ®X " then its image in IW52 i\ g equals zero. v

By 11.2 it suffices to show that its image in I~ [W = H°m(NzIV/ ' @Z ) is V

zero. Let • be a section of N~ i% V : since ~ is tangent to E at the smooth

points of 5~ ([, v> vanishes on an open dense set, thus ( ~, v> m 0 , ~ being

reduced. Q.E.D.

We shall not pursue here the analogue of Kuranishi's theory fo~ these

deformation theories (cf. [Sch], [Wahl, [B-W], [IDa]), in fact, as we have shown

already and will see in the sequel, it is very hard to compute the obstructions in

almost all the examples, whereas geometry can help to find a complete family of

deformations.

§12, Surfaces with ordinary singularities

Here X is a smooth surface and will hence be denoted by S, %o : S~ Z c W,

where W- is a smooth 3-fold is a finite map, birational onto its image ~, which

possesses only the following type of singularities :

i) nodal curve (xy = 0 in local holomorphic coordinates)

[i) triple points (xyz = 0 in local coordinates)

iii) pinch points (x Z - zy Z = 0 in local coordinates).

A will be the double curve (= Sing(Z)) of Z, smooth at points of type i), iii), with

local equations x = y = 0, and with a triple transversal point at each triple point.

We let D = %O-I(A) C S, and notice that a pinch point p' has just one inverse inaage

point p, where we can choose local coordinates (u,v) such that

(iZ. i) %0(u,v) = (uv, v, u z)

hence in particular D = [(u,v) Iv = 0 }.

Proposition IZ.2. If Z has ordinary singularities, the morphism %O is stable

l (i.e., ch. II.4, %O,(N0 ) --~ N EIW )"

Proof. In view of II. 6 and ii. 9, it suffices to consider the case of triple and pinch

9~ goes onto %O,. N . To do this, we shall explicitly points, and to prove that I V/ -,- %O

compute these two sheaves.

! Lemma IZ. B. NEIV/ c NEIV/ -- @E (Z) is the subsheaf of sections g vanishing

on & and satisfying the further linear condition: 0g/0y = 0 at the pinch points.

42

Proof. g E N' iff it belongs locally to the Jacobian ideal of E, i.e. (x,y) for EIW 2

the nodal poin ts , (xy, yz, xz) for the t r ip le points, (x, y , yz) fo r the pinch poin ts . Z

A t t he p i n c h p o i n t s , g E ( x , y , y z ) ~=> g : x g 1 +Yg2 with gz(O,O,O) = 0

a g / a y (0 ,0 ,0 ) = 0. Q.E.D.

l Remark iZ. 4. Since g vanishes on A , clearly ~g/Dz : 0 at a pinch point p .

Hence the condition 8g/~y : 0 can be formulated also as : ~ (g) = 0 for each tangent

vector at pt lying in the tangent cone to E, ~x Z : 0 ] . Clearly this last formulation

is independent of the choice of coordinates.

L__ernm_a. iZ____. 5. Let Pl' .... Pk be the points of S mapping to the pinch points k *

' ' ~ ?~Pi (9 s( %0 5" _ D). Pl ..... Pk of Z : then Nc# ~i:l

Proof. By II.9 and IZ. 3 we know that N ) coincides with %0*(N~ IV/) except at a

finite number of points, and that ~'~(N~ IW ) equals to t9S{%0 E- D), where is

an ideal sheaf of a 0-dimensional scheme. Hence N is also of the form N = %0 %0 J@S(%0* E-D), with dim(supp((gS/j) ) = 0. To determine the ideal J, we first notice

. . . . ~ coker @Z ~ @3 where that supp(@s/~9 ) = [PI' .,pk} then that, at Pi N%0

{~ = differential of %0, sends apair (gl,g2) to atriple fl :vgl +ugz' fz =gz '

f3 : ZUgl" The homomorphism of @3 -> ~Pi sending (fl,fz,f3) to (Zuf l-vf3)

clearly gives an isomorphism of N with ~Pi " Q.E.D.

l V/e can now finish the proof that <0 .~(N%0) : N E IV/ : in fact ~,(gs(-D) =

(gE(.A), as it is easy 9o see, whereas at the pinch points ?~PI' @s(-D) : (uv, vZ),

whereas, g E N Z' I V/ iff, g = xgl + YgZ with gz E ~p,i ; as we have already seen,

%0 (g) = uv %0*(gi) + v %0 gz E ~pi(gs(-D), and we can conclude since both sheaves

%0.(N%0) ~ NE IW have cod imens ion 1 in (gE( E- A).

Q.E.D. for Proposition iZ.Z.

Corollary IZ. 6. If E has ordinary singularities only in the smooth 3-fold V/, then

t h e r e exists an exact sequence

o * Ho(N~ W) -~ HI(~,S) -~ H1(%0* V/) 0 . H ° ( ~ S ) -~ H (%0 %V) -~ I

1 Z 2 • HZ(N~ IV/) -~ H (N Z IVy) -> H (~;S) -~ H (%0 ~W) -~ -~ 0

Proof. Obvious from the Leray spectral sequence for the finite map ~. Q.E.D.

43

Definition IZ. 7. @W (E) (- A, c'), where c' stands for "cuspidal conditions, " is

defined to be the inverse image of N' {W under the surjective homomorphism

~w(z) ~ ~z(z) = N z.

The heuristic explanation for (gW(E)(-A- c I) (cf. [Ko 2]) is as follows:

assume that you deform the singular locus of E by deforming with a parameter

the local coordinates x, y, z; then if X(t) = x + t~ + --. , Y(t) = y + trl + ....

Z(t) = z + t~ + ..., the local equation of ~ changes as follows:

XY = x y + t(gy + fix) + --.

XYZ =xyz +t(~yz + ~xz + ~xy) + ...

2 2 2 2 y2) X -Y Z =x - y z t(Zgx- 2zyq- ~ + .-.

Hence, if f = 0 was the old equation, the new one is of the form f + tg + ....

where g is a section of @W(E) vanishing on A and satisfying the cuspidal

conditions.

We clearly have an exact sequence (f is a section with div(f) = E) °

(12.8) 0 -~ ~W f + 0 %( z ) ( -~ - c ' ) - ~ N~: l w '

and Kodaira, after Severi, gives the following (cf. [Ko 2]).

Definition IZ. 9. ~ is said to be regular if HI(@w(~)(-A- c')) = 0, and semi-

regular if HI( @w( E)(-A- c t)) -~ HI(Nz IW ) is the zero map.

HI( 3 Remark 12.10. The two definitions coincide if {gW) = 0, e.g. for V£ = IP .

We h a v e the following.

Theorem ig. II (llodaira, [Ko 2]). If E is semi-regular the characteristic system

of the map ~0 : S -* • is complete; moreover, there is a smooth semi-universal

family [~Pt ] of deformations of q0 : S -~ ~ such that the characteristic system is

complete also for t ~ 0.

Unfortunately, the condition of semi-regularity is a very strong assumption

upon ~ c %V: we shall, following Kodaira ([Ko Z] , [Ko 3]), consider from now on

only the classical case where V/ = ~ D3 , and regularity coincides with semi-

regularity.

44

Theorem 12. 12. If E is a surface in [p3 of degree n with ordinary singularities,

52 is (semi)-regular if and only if the cuspidal conditions are independent on the

space of polynomials of degree n vanishing on the double curve A of 52 (i. e. ,

8g / Oy i 0 -~ H ° ( @ K ~ 3 ( n H ) ( - ~ - c t ) ) -~ H ° ( ~ 3(ni l ) ( -&) - ; • ~3 t -~ 0

[ P i

i s e x a c t , w h e r e H is the h y p e r p l a n e d i v i s o r on ~3 ) .

Proof. By assumption

HI((9 3(nH)(-f~- c') = lll(@[m3(nH)(-A))

By the exact sequence

0 -> @ip 3 -~ @o3(nH)(-A)-* (9~ (nH)-A) * 0

we have thus to show that HI((952(nH-£~)) = 0° Denoting still by H the pull-back of

a hyperplane, we have H1((952(nH-Z~)) = HI((gs(nI-I- D))° Since, by adjunction, the

canonical divisor on S is (n-4)H- D, by Serre duality, our space is dual to

HI(@s(-4H)), which is zero since H is ample (e.g. by Kodaira's vanishing theorem).

Q.E.D.

The preceding criterion of regularity is not so easy to apply directly,

thus the usual method is to relate the equisingular deformations of 52 to the

(equisingular) deformations of A (observing that sections of (9 3(nil) vanishing ~o

on A of order 2 give trivial infinitesimal deformations of A ).

We have the usual exact sequences (cf. 9. 17)

(12.13. i) 0 -> ® 3(-log A) -> ®fp3 -+ NA]ip 3' -> 0

®~3 (12. 13. ii) 0 -> ®A -> ® @A -~ NiXIe3 -> 0

and moreover ([l<o 3] , thin. 4).

Theorem IZ. 14. There exists an exact sequence

0 + @ 3(nH)(-ZA) -~ @ 3(nH)(-A-c t) -> N tAI~3 -~ 0 .

Idea of Proof (see loc. cir. for details). Let ~ be A- [triple points and pinch

points of 52 } , and let N be the normal bundle of ~ in ~3 . Since the conormal v

sheaf N of A is just @ 3(-A)/@ ~(-2A), the basic claim is that there exists an

isomorphism of N into N I~ ® @[p3(nH): and after that one has to check that this

isomorphism extends at the cuspidal points onto the subsheaf defined by the cuspidal

45

conditions, and at the triple points there is a similar verification. Since ~ and V

1'4 I~ are dual bundles, the key point is that the equation f of Z, locally of the v

form xy = 0, induces locally two sections of N I~, and globally a non vanishing 2 v

section of A N I~ ® @ 3(nil), thereby inducing a non degenerate pairing

× ~l -> (9~(nH), hence the desired isomorphism. Q.E.D.

The important feature of IZ. 14 is that the left term of the exact sequence

depends only upon the double curve A and the degree n of E , but not upon 5~.

Moreover, given any curve A in IP 3, by Serre's theorem ([Se]), there is an

integer

(12.15) no(A) = rain [nlHi((9 3(kH)(-ZZN) = 0 V i= 1,2, k mn] @

Theorem iZ. 16. Let X be a surface of degree n with ordinary singularities in ~3

having A as double curve; if n ~ n CA), 7. is regular if and only if HI(N ' ip3) = 0. o &I In particular ~ is regular if Hl(® 3 ® (gA) = 0.

Proof. By the cohomology sequence attached to IZ. 14, and by iZ. 15,

H I _ ~_ HI(iN t A (® 3(nH)C-a c%) [~3).

The other assertion follows from (IZ. 13. ii). Q.E.D.

Theorenq IZ. 17. Let Z be a surface of degree n with ordinary singularities in ~3

having A as double curve, and let 7k be the normalization of A.

i) If T is the divisor on ~x given by the sum of the triple points, and H is

the hyperplane divisor, then if n ~ noCA) and @~(H-T) is non-special

on Cevery component of) ~x, then Z is regular.

ii) If there exists a surface Z l of degree n' containing A, and such that

the divisor ~;;"(Z I) on S has no multiple components, then

n (Z~) ~ n +n'- 3. o

Proof. i) by 1Z. 16 it suffices to show HI(® 3 ® (9^) = 0. By the Euler sequence

(6.3) tensoredwith @A' it suffices to showthat HI(@AC1)) = 0. Now, if @: A-> A

is the normalization map, ~, (@~(H-T)) = F~T @A CI)' where ~T is the ideal sheaf

of the triple points. Hence HI(A T @AC1)) = 0 and we are done by the exact sequence

0 -> ~T (%A (I) -> (gACI) -> T -> 0

where T is a skyscraper sheaf with stalk -~ C at each triple point.

46

ii) let k be an integer a n +n' - 3 and consider the exact sequence

0 -~ @ 3[kH-E) ~ (~[o3(kH)(-Z~)-~ (gE(kH-ZA) -~ 0

Since Hi(@ 3(kH-E)) = Hi(@o3((k-n)H)) = 0 for i= 1,2, it suffices to show the

vanishing o~ HI(@E0cH- ZA)). Since @E(-ZA) = ~:. @s(-ZD), we want the vanishing

of HI(@s~H-ZD))° As in iZ. IZ, since K S = (n-4)H- D, the dual vector space is, by

Serre duality, HI(@s (- (k-n÷4) H+D)). }By assumption, nIH m q~;:"(Z I) = D + F,

HI(@s(-aH - F)), where a--k-n+4-n ta i. But hence we w~nt the vanishing of

l all + F I maps to a surface and faN + F I contains a reduced connected divisor,

hence one can apply the Rarnanujam vanishing theorem (cf. e.g. [Bo] ,tRam) ).

Q.E.D.

By IZ. 14, if n ~n (A), then I-I°(@ ~(nH)(-f~-c')) goes onto H°(N ' 3): o [p~ ZXiiP

on the other hand, by (IZ.8) this surjective homomorphism factors through fhe one

onto H°(N~. lip3), which has the subspace (Ef as its kernel (f = 0 being the equa-

tion of E). Assume now /~ to be smooth (thus E has no triple points): then if the

characteristic system of ~< is complete, and n ~ no(A), then also the characteristic

system of A is complete; moreover, Kodaira (loc. cir., p. Z46) proves the

converse.

Theorem IZ. 18. Let ~ be a surface of degree n in R ~3 with ordinary singulari-

ties and smooth double curve A. Assume n ~ n (Z~): then the characteristic sys- o

tem of A is complete if and only if the characteristic system of E is complete.

This theorem, combined with Murnford's example 9. II of a family of space

curves f~ for which the characteristic system is never complete for each f~, shows

the existence of many surfaces ~ such that all their equisingular deformations do

not have a complete characteristic system: in fact, given an n such that

@iim3(n)(-Zf~) is generated by global sections, it follows by Bertini's theorem that

the general section f E HO(@D3(n)(-ZA)) defines a surface Z smooth away from A,

and with ordinary singularities only.

This result, obtained Z0 years ago, culminated a very long history of

attempts to show that the characteristic system of a surface E with ordinary singu-

larities should always be complete (we defer the reader to ten), [Za], especially

Munnford's appendix to chapter V for a more thorough discussion).

47

We simply want to remark again that the fact that the characteristic system

is not corn plete does not imply the singularity of the base B of the Kuranishi

family: in fact, for t E B one can have a deformation q0t~of the holomorphic map

: S -> IP 3 if and only if the cohomology class of H (= ~'" Oayperplane)) remains

of type (i, i) on S t .

Example iZ. 19. A classical case where Kodaira's theorem 12.17 applies is the

case of Enriques' surfaces Z, with equation

3

0 i+0 i=O x .

where q(x) is a general quadratic form. Here ~ consists of the six edges of the

coordinate %etrahedron [X0XlXzX 3 = 0 } , n = 6. The normalization ~ of A con-

of 6 copies of [pl , and (%~(H-T) has degree (-l) on each component, hence sists

is non special (HI((qRDI(-I)) = 0!). The surface 51' to be taken is a general cubic

surface of equation

3

X0XlXzX 3 a i : 0 , 0 ~

hence no(A ) ~ 6, and Z is regular.

The characteristic system has dimension Z5 and it is easy to see that a

smooth complete family of deformations of E is obtained by taking images under

projectivities of surfaces in the above 10-dimensional family. Working out the exact

sequence 12.6, we see that the above 10-dimensional family has bijective Kodaira-

~pencer map, so that the Kuranishi family of S is smooth, 10-dimensional. This

last result can also be gotten in a simpler way: since K S = ZH - D, and # (X0XlXzX3) = ZD, we get ZI4 S m 0. On the other hand, if K S 0, there would be

a quadric containing A, what is easily seen not to occur. Hence K S ~ 0, ZK S -~ 0 ,

Moreover X(@S) = i. Taking the square root w of X0XlXzX 3 and then normalizing

the surface Z t = [(w,x0,xl,Xz,X3) lw Z = xoXlXzX3 , f(x) = 0} , we get a smooth

surface S' (called a I<3 surface) possessing an unramified double cover II : S' -> S. o 1

Iris easyto see that 1<S' ~- 0, and, since $(((9S, ) = Z, HI((gS ,) = H (QS,) = 0 . o I ~2 i

Now HZ(®s) is the Serre dual of H (~S ® S t) = H°( ~S t ) = 0, hence there are

no obstructions for the Kuranishi family of S (of S s , too).

48

Example lZo20 ([Ko 3], [Hor 4] , [Us ]). Let ZX c ~3 be a smooth curve , the

complete intersection of two surfaces, A = iI ~ = G = 0 } . Then one can consider

the smooth fan%ily of surfaces of degree n having ~ as double curve. ]By our

assumption, it is easy to check that, if deg F = a, deg G = b, the equation of E

can be written in the form

(IZ. ZI) AE Z 4 ZBFG + CG Z ,

where deg A = n-Za, deg B = n- a-b, deg C = n- Zb.

The results of I<odaira-Horikawa and Usui can be summarized as follows :

varying A, B, C one gets a surjective characteristic map, so that

(1Z. Z2 i) the c h a r a c t e r i s t i c s y s t e m is comple te .

Using the s tandard Euler sequence , it is poss ib le to p rove that , in the exac t

sequence

H1 N°(IV'ZI~ 3) -~ HI(~s ) ~ (~* ®~3) -~ ~t l (N'zI~ 3)

the h o m o m o r p h i s m c; is in jec t ive , hence in p a r t i c u l a r

(12. ZZ i[) The Kuranishi family of S is smooth.

(12.22 iii)

(12. Z2 iv)

Furthe rmo re,

the above surfaces are not (semi)-regular,

the pairing HI(®) × H°(f1~) -~ }Ii(f~%) is non degenerate in the first

factor (this result is called Infinitesimal Torelli property, and actually

Usui proves the above result, provided n ~ n (A), also in the more o

general case of theorem 1Z. 17).

This example shows clearly how the condition of semi-regularity is much

too restrictive (in fact, as we noticed, it is an analogue of the condition HZ(®X) = 0

in order to ensure smoothness of the Kuranishi family).

§13. Generic multiple planes and equisingular deformations of plane curves with nodes and cusps

We consider again a smooth surface S and a finite morphism q0 : S -> 2 ,

of degree d; we let, as usual, H be the pull-back of a line, and we denote by R the

r ami f i ca t ion d iv i so r of ¢p, i . e . , g iven the exac t sequence

49

®S ®~2 -~ 0 (13.1) 0 -~ • q0 -~ N

2 2 t h e d i v i s o r of z e r o s of (A ~ , ) E H ° ( ( A

The f o l l o w i n g is a c l a s s i c a l

;:-- Z v

® z) ® (A ®s) ).

D e f i n i t i o n 1 3 . 2 . ~P is s a i d to be a g e n e r i c m u l t i p l e p l a n e (or a s t a b l e m o r p h i s m ) i f

R i s s m o o t h , q0(R) = B ( the b r a n c h l o c u s ) h a s o n l y n o d e s and o r d i n a r y c u s p s a s

s i n g u l a r i t i e s , cO IR : R -~ B is g e n e r i c a l l y 1 -1 . L

At t h e p o i n t s of R , t he n o r m a l f o r m of %0 ( fo r s u i t a b l e l o c a l h o l o m o r p h i c

c o o r d i n a t e s in t h e s o u r c e a n d i n t he t a r g e t ) i s as f o l l o w s :

(13.3 i)

(~ 3.3 il)

~p(x,y) : (xZ,y) at the points p £ R with ~0(p) not a cusp of B

t %0(x,y) = (y, yx-x 3) at the points Pi of i% with %0(pi) = Pi a

cuspidal point of B.

(13.3 i) and ii) imply that N%0 is an invertible @R-sheaf. It is easier though, to

compute q0~:(Ncp): we apply q0 to 13. 1 to obtain

0 -* ~, ®S ~ ®~2 ® cp.~ @S -~ %0*Ncp

Proposition 13.4. The trace map t: ®~2 ® %0* @S -~ ®lp z i n d u c e s an i s o m o r p h i s m

®~Z / ®~2 -~ : N' of q0 Nq0 with (-log B); in particular, ~0 N%0 BI 2 .

Proof. The statement is almost obvious for the points of [pZ which are not cuspidal I J

points of B. At a cuspidal point Pi ' if Pi E R is such that q?(Pi ) = Pi ' we see

immediately that

(CP '~ N ¢9)P'i " "" ' P i

T h e s e two a r e @ 2 , m o d u l e s a n d , i f w e c h o o s e c o o r d i n a t e s a s in I 3 . 3 i i ) , [P 'Pi 3

(y,z) at p{ such that z = yx - x a basis for q0 • is given by ' S , Pi

B B 2 B B B 2 B Bx , x ~x ' x Bx " By ' x , x By '

w h i l e a b a s i s f o r 8 Z ® %0~ @S, p i i s g i v e n b y

8 8 Z 8 B 8 28 B y ' x , x B y ' Bz ' x ~ , x 8z

50

(0 i) S i n c e t h e J a c o b i a n m a t r i x <P.. i s , w e s e e t h a t t h e i m a g e o f $ i s t h e

" ( y - 3 x Z) x s u b r n o d u l e g e n e r a t e d b y

0 0 O 2 0 2 0 0 ~ + x ~ , ~ 5 ~ + ~ ~ , x ~+(r~-~)~,

( i 3 . 5 ) 0 0 8

( y - 3 x 2 ) ~ z ' ( 3 z - Z x y ) ~zz ' ( 3 x z - Zyx z ) ~ z "

W e u s e n o w t h e s y m b o l = to d e n o t e c o n g r u e n c e m o d u l o t h e s u b m o d u l e i m a g e ( ~ ) ,

a n d w e d e d u c e f r o m ( 1 3 . 5 ) t h a t

8 = 8 2 0 1 O 8 2 0 1 8 o~ Oy ~ = . ~ Y ~ z ' x ~y --- o~,

2 3 0 0 0 0 0 ~_ 0 8__ x ~ y -= z ~z " y x ~ z ~ z ~z + y ~ y ' 3z ~zz 2 y x ~ z ~ - 2 y 8y '

0 2 y x 2 ~ _ 0 2 2 8 0 ~ 3 z x ~ZZ " Oz - 3z ~yy - - ~ y O--z "

We readily infer that <01 = (N~) is isomorphic to the quotient of ® Z by the sub-

2 ~ module generated by 2y(8/0y) + 3z(8/0z) and by 9z(8/0y) + 2y (0/Oz): it is immed-

iate to check that this last submodule is indeed ®[pz(-log B), since the equation of

B = ~(R) is given by 4y 3 - ZTz 2 = 0. Q.E.D.

The previous proposition shows that infinitesimal deformations of the stable

map ~ correspond to infinitesimal equfsingular deformations of B. On the other

hand, if [~t: St ~ [p2 }t E T is a deformation of ~0, it is easy to verify that the

condition that ~t be stable is an open one, and [Bt} = [[p~(Rt) } is an equisingu-

lar family of plane curves with nodes and cusps. Conversely, if [Bt }t E T is an

equisingular family of curves with nodes and cusps which is a deformation of B = Bto,

we see that for t near to to the pairs (IP Z, ]3) and (~ DZ, B t) are diffeomorphic,

and in particular [II([D2-Bt) ~ [II(~2-B). Thus, the associated subgroup of the

covering %0: S- ~-I(B) -> [DZ-B determines another smooth surface S t with a

stable morphism ~t: St -~ p2 and it is not difficult to verify that in this way we get

a deformation of ~ with base T° We have thus

Theorem 13.6. There i s a natural isomorphism between the characteristic system

H°(N ) of a generic multiple plane %0 and the equisingular characteristic system

H°(N' B ~2) of its branch curve B. Also, the characteristic system of %0 is com-

plete ~ and only if the equisLngular characteristic system of B is complete.

51

A g a i n 9 . 1 1 and 1 2 . 1 8 i m p l y the e x i s t e n c e of p l a n e c u r v e s w i t h n o d e s and

cusps whose equisingular characteristic system is obstructed: in fact, if

~0' : S -> Z c 3 is a map to a surface with ordinary singularities, it is well known 3 2

that there exists a point p in ~p3 such that the projection I~ : - [p } -~ with

centre p makes q0 = II o opt into a generic multiple plane. It is then clear that if

i s a d e f o r m a t i o n o f then is adefor atio of Con-

v e r s e l y , as remarked before example 12.19, if ~ ~t ] is a deformation of t , then

there is a deformation ¢0 ' of cO' if and only if setting H t = cp* 0xyperplane in IP2), t t

the four sections x 0 ..... x 3 of H°(@s(H)) extend holomorphieally in t to 4 sec- o

tions x0t ..... x3t of H (@st(Hi)).

This property holds in particular if dim HI((gst(Ht)) is independent of t

we defer the reader to [Wah] for more details, as well as for a very precise

account of the theory of equisingular deformations of curves with nodes and cusps.

Again here we have to remark that Enriques tried several times to show that curves

with nodes and cusps were unobstructed, but this is not true, by the example of

M u m f o rd-Kod a i r a - Wahl.

52

LECTURE FIVE: SURFACES AND THEIR INVARIANTS

~14~ Topological invariants of surfaces

In this section and the following ones, we shall very quickly review some

basic facts about the topology of compact complex surfaces, and roughly outline

the Enriques-Kodaira classification of (compact) complex surfaces. We defer the

reader to [Bo-Hu], [Be I], lIB-P-V], and also to the survey papers [Ci], [Ca 2]

for a thorough, update and exhaustive treatment. Given a (compact) complex sur-

face S, we shall consider its underlying structure as an oriented topological 4-

manifold, and also its differentiable manifold structure.

The main topological invariants of S are

I~I(S): the fundamental group of S

b.(S)i = dim~.~ Hi(S,~), the Betti numbers of S 4

e(S) = Z (-i) ib (S) = Z - 2b +b 2 the topological Euler-Poincar~ i=0 i i '

characteristic of S

T: the torsion subgroup in HI(S,~) (and in H2(S,~)) (14. 1)

Q = H2(S,~) -~ ~, the integral unimodular quadratic form given by

cup product (composed with evaluation on the fundamental

class of S)

b+,b-: the indices of positivity, resp. negativity, of Q + -

= b -b : the signature of the manifold (note that the rank of O

is b 2 = b + + b-)

The differentiable structure determines the real tangent bundle of S and its second

Stiefel-Whitney class w2(S) (cf. [Mi-Sta]), by a theorem of Wu, determines wheth-

er Q(x) is an even (i.e. Q(x) even V x in HZ(S,2~)) or odd form, since Q(x) -~

wz(S) • x (rood 2). Now, it is known (cf. [Se 2] ) that all indefinite unimodular quad-

ratic forms are determined by their rank, signature and party: if they are odd,

then they are diagonalizable over 2~ (hence with ± 1 entries on the diagonal), and

if they are even, they can be brought to a block diagonal form, with building blocks

u2 (0 l) and 0 2 0 -I O'

/ 0 2 0 -I ', l-i 0 2 i :

E 8 = ,| 0 -I -I Z ,-I or -E 8 . . . . . . . . - i - z - 1

- I 2 -1 -1 Z

53

N o t i c e t h a t ~'(U2) = 0, ~r(E8) = 8 , and , by a t h e o r e m of R o k h l i n i r k ] , if w z = 0,

t h e n ~ - 0 (rood 16) (w Z = 0 ~ Q is e v e n , bu t no t c o n v e r s e l y , e l . [Hah] o r 17, 6).

What c a n be s a i d a b o u t Q w h e n Q i s d e f i n i t e ? D o n a l d s o n [Do 1] r e c e n t l y

e s t a b l i s h e d the f o l l o w i n g r e m a r k a b l e r e s u l t .

T h e o r e m 1 4 . 2 . L e t M be a c o m p a c t o r i e n t e d 4 - d i m e n s i o n a l m a n i f o l d wi th d e f i n i t e

i n t e r s e c t i o n f o r m Q: t h e n Q is d i a g o n a l i z a b l e (i. e . , i t s m a t r i x is :5 I d e n t i t y in a

s u i t a b l e b a s i s ).

The i m p o r t a n c e of the i n t e r s e c t i o n f o r m Q l i e s in the f a c t t h a t i t i s the

un ique t o p o l o g i c a l i n v a r i a n t w h e n the 4 - m a n i f o l d M is s i m p l y c o n n e c t e d . We h a v e

in fact ([Fre])

T h e o r e m 14 .3 (M. F r e e d m a n ) . L e t M, M ' be c o m p a c t o r i e n t e d t o p o l o g i c a l 4-

m a n i f o l d s , and a s s u m e t h a t t h e y a r e s l m p l y - c o n n e c t e d , and h a v e the s a m e i n t e r -

s e c t i o n f o r m Q. If M, M ' h a v e a d i f f e r e n t i a b l e s t r u c t u r e , t h e y a r e t o p o l o g i c a l l y

e q u i v a l e n t . M o r e g e n e r a l l y , g i v e n Q, t h e r e a r e a t m o s t two t o p o l o g i c a l t y p e s of

4 - m a n i f o l d s M wi th f o r m Q and [I I (M) = 0: if t h e r e a r e two, t h e y a r e d i s t i n -

g u i s h e d by the p r o p e r t y w h e t h e r M × [0, 1] a d m i t s o r d o e s n ' t a d m i t a d i f f e r e n t i a b l e

s t r uc t u r e.

§ 15. A n a l y t i c i n v a r i a n t s of s u r f a c e s

B e f o r e g iv ing a l i s t of i n v a r i a n t s , i t i s c o n v e n i e n t to c l a r i f y t h a t in s o m e

c a s e s we a r e t a l k i n g a b o u t b i h o l o m o r p h i c i n v a r i a n t s , in o t h e r s a b o u t b i m e r o m o r p h i c

i n v a r i a n t s . To e x p l a i n the n o t i o n of a b i m e r o m o r p h i e m a p , we r e c a l l t h a t two

s m o o t h a l g e b r a i c v a r i e t i e s X a n d Y w e r e c l a s s i c a l l y s a i d to be b i r a t i o n a l if

t h e i r f i e l d s of r a t i o n a l f u n c t i o n s ~ (X) , ~E(Y) w o u l d be i s o m o r p h i c . Such an i s o -

m o r p h i s m d o e s not i nduce a b i h o l o m o r p h i c m a p , but on ly a " g e n e r a l i z e d " g r a p h ,

i. e. , a closed subvariety F of XXY such that, Pl' P2 being the projections on

b o t h factors:

(15. 1) i)

ii)

t h e r e e x i s t c l o s e d s u b v a r i t i e s I X of c o d i m e n s i o n a t l e a s t 2 in X

( r e s p . : I y ) , s u c h t h a t the r e s t r i c t i o n of Pl f r o m

F - p l l ( I x ) -~ X - I X

is b i h o l o m o r p h i c ( s a m e c o n d i t i o n fo r p2),

F is irreducible (hence F is the closure of F- pil(Ix)).

54

Replacing the word "subvariety ~' by the word "closed analytic subspace" (actually,

in the sequel we shall make little distinction, since by Chow's theorem a closed

analytic subspace of a compact algebraic variety is an algebraic subvariety), we

obtain the definition of a bime romorphic map.

Definition 15.2. A bimeromorphic map between compact complex manifolds X, Y

is a biholomorphic map %0 between open sets Of X and Y, U X and Uy , such

that X- U X , Y - Uy are closed analytic subsets, and the closure F of the graph

of %0 is a closed analytic subset of X xY satisfying properties 15. i.

In general, for a compact complex manifold X, we denote by ~(X) its field of

meromorphic functions and we recall the following famous result of Siegel [Sie i].

Theorem 15.3. C(X) is a finitely generated extension field of (D with transcend-

ence degree a(X) over ~ with a(X) ~ n = dirn~ X°

Remark 15.4. I£ is easy to see that abimerornorphie map between X and Y in-

duces an isomorphism between (g(X) and ~7(Y). Moreover, for an algebraic variety X

C(X) coincides with the field of rational functions and, if dim Y - a(Y), ~(Y) ~-

C(X), then Y is bimerornorphic to X. Such a Y does not need to be algebraic if

dim a 3, and is usually called a k~oishezon manifold (el. [Moi i], [Moi Z]).

In dimension Z the bimeromorphic maps are obtained as composition of certain

elementary birneromorphic maps which we are going now to describe.

Example 15.5. Let X be a complex manifold, p a point in X, (z 1 ..... Zn) coordi-

nates in a neighborhood U of p, with p corresponding to the origin. Let ~7 be

the closure of the graph of the meromorphic map U -* ~ on-1 sending (z 1 ..... Zn) to

the line (E(z 1 ..... Zn). Then, glueing ~7 with X- [p} in an obvious way, we ob-

tain a new manifold X, with a proper holornorphic and bimeromorphlc map

C;: X-> X such that

i) G-l(p), which is called the exceptional subvariety and denoted by E, is

isomorphic to ~ °n-1

i[) The normal bundle NEI ~ is isomorphic to @Kon_l(-l).

iii) C; I~_E is a biholomorphism.

¢r is called an elementary modification and C;-1 is called the blow-up of the point

P.

55

T h e f o l l o w i n g r e s u l t is c l a s s i c a l .

Theorem 15.6. Everybimeromorphic map of complex surfaces factors as a com-

position of blow-ups followed by a composition of elementary modifications.

Definition 15.7. A complex surface S is said to be minimal if every holomorphic

and bh-neromorphic map G : S -* S' is a biholomorphism.

Remark 15.8. If S ~ S is an elementary modification, then the lattice HZ(s, ~),

equipped with the quadratic form Q, is the orthogonal direct sum HZ(S, 2) @ ZE,

where E is the cohornology class of the exceptional curve E, with Q(E) = -i. In

particular Q is odd and bz(S ) = b2(S ) + I.

Conversely, if a complex surface S contains an exceptional curve E of the

I k i n d , i . e . E - - [pl , NE1 ~ = @ i ( - i ) ( o r , e q u i v a l e n t l y Q(E) = - i ) t h e n t h e r e

e x i s t s a n e l e m e n t a r y m o d i f i c a t i o n ~ : S -+ S w i t h ~(E) = a p o i n t in P , a n d the

s e c o n d B e t t i n u m b e r of S is e q u a l to b z ( S ) - i . T h i s is t he c l a s s i c a l r e s u l t of

C a s t e l n u o v o a n d E n r i q u e s , e x t e n d e d b y X o d a i r a in the n o n a l g e b r a i c c a s e , a n d g e n -

e r a l i z e d b y G r a u e r t in [ G r Z] ; c o m b i n i n g t h i s w i t h a n o t h e r d e e p t h e o r e m of

C a s t e l n u o v o and K o d a i r a , we o b t a i n the f o l l o w i n g

Theorem 15.9. A complex surface S is minimal if and only if it does not contain

an exceptional curve of the I kind. Every complex surface S' is a blow-up of a

minimal surface S, and S is unique up to biholomorphism except if S' is ruled,

i.e. S' is bimeromorphic to a product CX~ °l .

Another biproduct of the structure theorem 15.6 of bimeromorphic maps of

surfaces is the following theorem of Chow and Kodaira.

T h e o r e m 15.10 . A ( s m o o t h ) c o m p l e x s u r f a c e S is p r o j e c t i v e ( i . e . , a s u b m a n i f o l d

of s o m e p r o j e c t i v e s p a c e ) i f a n d o n l y i f a(S) = Z.

We s h o u l d a l s o r e m a r k t h a t , b y t he r e s u l t of K o d M r a ( [Ko 1 ] , t h i n . 6)

q u o t e d a t t h e e n d of §9, a l l s m a l l d e f o r m a t i o n s of a n o n m i n i m a l c o m p l e x s u r f a c e

a r e a g a i n n o n m i n i m a l , w h i l e we s a w (use p r o p . 6 .19 , a n d t he f a c t t h a t ~ 1 i s t he

b l o w - u p of a p o i n t in {pZ) t h a t i s is n o t t r u e f o r d e f o r m a t i o n s in t he l a r g e .

We c a n now s t a r t to r e v i e w s o m e of the c l a s s i c a l b i m e r o m o r p h i c i n v a r i a n t s

of s u r f a c e s . T h e f o l l o w i n g a r e n u m e r i c a l i n v a r i a n t s .

56

(15. ll)

pg , the ge,gmetric genus, is h2(@ S) = d[mensioncg H2(@S )

q , the i,r,re~ularity, is hl((9 S) ® m

Pro' the mth plurigenus, is h°((f~2) ) = h°((gs(mK)

A more subtle invariant is the graded ring

(15.1z) ~(s) =

co

@

m=0 H°(@s(mK)), the canonical ring.

Definition 15. 13. Let Q(S) be the field of fractions of homogeneous elements of the

same positive degree in ~(S): then, either Q(S) = ~, or Q(S) is algebraically closed

in ~(S), and the Kodaira dimension of S, Kod(S) is

{ -~ if O(S) = ¢

tr deg C Q(S), otherwise.

th The above definition is not the unique possible one : denoting by ~0 the m

m

pluricanonical map, i.e. the rational map ¢Pm: S .... > ~ Pm'l attached to the O

sections of H (@s(mK)), one can also define Kod(S) robe max (dim~m(S)). m

From the complex point of view, one can consider divisors D, D' and con-

sider the intersection number of t~o divisors, D • D' , as classically defined

through algebraic equivalence: since to a divisor D one associates the invertible

sheaf @s(D), one sees that, denoting by c I (I Chern class) the homomorphism of

HI(@s) (classifying isomorphism classes of invertible sheaves) to ~) appear- H2(S,

ing in the cohomology sequence of the exponential sequence

0 -~ 2rTi~ -~ (9 exp~ (9~ -~ 0 ,

the bilinear form of intersection on H2(S, 2~) extends the classical intersection

product. It is in fact true more generally that many of the analytical invariants,

whose definition depends upon the complex structure of S, are in fact determined

only by the topological structure of S .

Notice that, by 15.6 and 15.8, Ill(S), T, b l, b + are bimeromorphic

invariants.

Another classical invariant is

(15. 14) p(1) K Z = + i, the linear genus of S.

KZ p(l), • as well as are not bimeromorphic invariants, but, if a surface S' is

not ruled, one can consider, e,g. , the linear genus, and all the possible analytical

57

and topological invariants of the unique minimal surface S bimeromorphic to S'

(S is called the minimal model of S' )o

If you perform ablow-up, e = 2-2b l+b Z goes up by I, K Z drops by I:

hence K Z + e is a bimeromorphic invariant, and an extension to complex surfaces

of a classical theorem of Noether identifies it with a combination of previously

encountered invariants. We have (cf. [Hi 1] ), the following

- pg) - is the Euler- Theorem 15. 15. (K 2 + e) = IZ(l q + = IZ X (X = i q +pg

Poincar~ characteristic of the structure sheaf (~S).

A n o t h e r r e s u l t of t h e s a m e t y p e , w h i c h r e p r e s e n t e d a r e a l b r e a k t h r o u g h in

t he c l a s s i f i c a t i o n of c o m p l e x m a n i f o l d s , i s t he i n d e x t h e o r e m of A t i y a h - S i n g e r -

H i r z e b r u c h .

Theorem 15.16. 3T = 3(b +- b-) = K z - Ze .

L e t ' s o b s e r v e now t h a t , b y S e r r e d u a l i t y , pg is a l s o the d i m e n s i o n of

HO(f~2 ); in g e n e r a l h o l o m o r p h i c 1 a n d 2 f o r m s on a s u r f a c e a r e d - c l o s e d , so t h a t S

t h e r e i s , u s i n g D e R h a m ' s t h e o r e m , an i n c l u s i o n H O ( f ~ l ) c H I ( s , •) ,

H°C~ ~ HZ~s,~). Moreover, if ~ ~H°(~S), ~ a ~-elosed (O,l) form andgives,

by Dolbeault's theorem, a cohomology class in Hl(@). In this way one sees that

b +>- 2pg, h ° ( Q S } ~ q , ( 2 q - b l ) ~ O, and the u p s h o t is t h a t , by a c l e v e r m a n i p u l a -

t i o n , the index theorem tells you that the sum (of positive integers!) (b + - 2pg) +

(Zq - bi) equals I: we thus have, (cf. [Ko 4]).

T h e o r e m 1 5 . 1 7 . If b 1 i s e v e n , b 1 = Zq, b + = Zpg + I , h ° ( f ~ l ) = q; ff b l i s odd ,

b I = Z q - 1, b + = 2 p g , h O { f ~ l ) = q - 1. In p a r t i c u l a r , i f b 1 i s e v e n a n d pg = 0, S

i s a p r o j e c t i v e s u r f a c e .

T h e r e a r e s e v e r a l o t h e r r e s u l t s p e r t a i n i n g to i n e q u a l i t i e s b e t w e e n n u m e r i c a l

i n v a r i a n t s , o r l i m i t a t i o n s in t h e i r r a n g e , b u t i t is m o r e c o n v e n i e n t to p o s t p o n e t h e s e

to t h e n e x t s e c t i o n , a s b e i n g p a r t of the c l a s s i f i c a t i o n t h e o r y .

We j u s t e n d t h i s p a r a g r a p h by m e n t i o n i n g a c o n s e q u e n c e of t he p r e v i o u s

t h e o r e m

C o r o l l a r y 1 5 . 1 8 . T h e i n t e r s e c t i o n f o r m of a c o m p l e x s u r f a c e S is s e m i - n e g a t i v e

d e f i n i t e i f a n d o n l y i f b 1 i s odd a n d pg = 0 .

58

LECTURE SIX: OUTLINE OF THE ENRIQUES-I<ODAIRA CLASSIFICATION

§16. Definition of the main classes of the classification

First of all, the main purpose of the Enriques-Kodaira classification is to

partition all surfaces, considered only up to bimeromorphic equivalence, into 7

classes, in such a way that the knowledge ~by explicit calculationsl of some numeri-

cal invariant (or some more refined invariant, as the triviality of the canonical

divisor) may allow to draw several conclusions about the structure and the geometry

of some surfaces taken into consideration.

Its analogue in dimension 1 is the rough subdivision of curves according to

the Kodaira-dimension, Kod = - o= if the genus g is 0, io eo the curve is R ml ,

Kod = 0 if g = i, i.e. youhave an elliptic curve, l<od = 1 if the genus is at least Z.

The classification of curves according to their genus is more refined, but it is

closely related to the knowledge of the topology of algebraic curves; in the surface

case, a complete classification, less rough than the one given by F_~nriques and

Kodaira, seems for the time being out of reach, due to the problem of classifying

all the surfaces of general type.

Definition 16. I. A surface S is said to be of general type if Rod(S) = Z, or,

equivalently, if Q(S) = ~($1 (cf. 15. 131.

Z Definition 16. Z. A surface S is said to be rational if it is bimeromorphic to

in particular a rational surface is ruled.

Definition 16.3. A surface S is said to be elliptic if it admits an elliptic fibration,

i,e., a surjective morphism f: S ~ B, with B a curve, and with the smooth

fibres of f being elliptic curves.

Remark 16.4. If S is elliptic, a(S) ~ a(B) = I; conversely, if a(S) = i, there

exists a curve B with •(S) m C(B), and an elliptic fibration f : S -~ B, such that

all the curves of S are components of the fibres of f. If S is elliptic, then

Rod(S) < I, and S is not of general type.

We can now pass to the list of the seven classes of surfaces, some of them

being divided into subclasses.

Class 1 I: Ruled surfaces (i.e. bimeromorphic to a product C × ~ml): these are

distinguished by the irregularity q which equals the genus of C, and

59

they are rational if q = 0. Their minimal models are [pZ • and [pl_

bundles over curves C.

From now on, since the minimal model is unique, we shall only talk about

the minimal models.

Class Z):

Class 3):

K3 surfaces, defined by the condition q = 0, K - 0°

Complex tori, i.e. surfaces biholornorphic to a quotient ~72/f~, where

f~ is a subgroup generated by 4 vectors linearly independent over ~.

Class 4): Elliptic surfaces with b I even, iOlZ # 0, K ~ 0, divided into two sub-

classes distinguished by the Kodaira dimension.

Class 4), Kod = 0: Enriques surfaces, i.e. normalization of surfaces as in IZ, 19,

or hyperelliptlc surfaces (explicitly described in [B-DF] , cf. [Be i] ,

pgs. llg-l14), quotients of a product of two elliptic curves EIXE 2 by

the action of a subgroup G of E 1 acting on EI×E Z by sending (x I, x 2) to

(xl+g, g(xz)), for a suitable action of G on E 2 such that E2/G -~ ~i .

Class 4), Kod = I: Canonically elliptic surfaces with b I even: ~01Z , the IZ th

pluricanonical map, gives an elliptic fibration.

Class 5) Surfaces of general type (cf. lecture seven).

Class 6) Elliptic surfaces wlth b I odd, l°iz ~ 0, with subclasses

Class 6), Kod = 0: Kodaira surfaces, i.e. surfaces of the form CZ/G, where G

is a group of a/fine transformations of the form (Zl,Z2)

(z I + a, z Z + ~ z I + b). They are distinguished into primary ones,

with b I = 3, K ~ 0, and secondary ones, with b I = l, K ~ 0; the

secondary ones admit an unramified cover of finite degree which is a

primary Kodalra surface.

Class 6), l~od = i: Canonically elliptic surfaces with b

elliptic fibration.

1 odd: ~12 gives an

Class 7): Surfaces with b I = I, PlZ = 0 (and Kod = -~ in fact): we shall not say

much about these, since their classification has not been yet complete-

ly accomplished.

60

§17. Criteria of classification and features of some classes

The following result is the prototype of all criteria of classification

Theorem 17. 1 (Castelnuovo's criterion). A surface S is rational if and only if

q=P2=0°

Ruled surfaces are also the ones which admit several characterizations.

even is ruled if and only if one of the follow- Theorem 17.2. A surface S with b 1

ing equivalent conditions is satisfied:

i) Plz = 0 J

ii) These exists a curve C, not exceptional of the I kind, with 1< • C < 0.

Moreover, if S is a minimal model and b I is even,

iii) K 2 < 0 if and only if S is ruled and q a 2 .

The following results hold instead when S is minimal, but without the assumption

that b be even. i

iv) K 2 > 0, P2 = 0 if and only if S is rational

v) I< 2 > 0, P2 ~ 0 if and only if S is of general type.

vi) e < 0 if and only if S is ruled and q ~ Z .

The definition of 1<3 surfaces we gave in §16 was one of the less explicit

ones : in fact, we could have chosen to define a 1<3 surface according to the follow-

hag beautiful theorem of Kodaira (conjectured earlier by Andreotti and %Veil).

Theorem 17.3. A minimal surface S is a 1<3 surface if and only if S is a direct

deformation, with non singular base, of a non singular surface of degree 4 in ~3 •

We notice that some of the I<3 surfaces can be elliptic, as well as some com-

plex tort, and this fact justifies the condition 1< ~ 0 used to define Class 4). We

defer the reader to [B-P-V] for an excellent survey about 1<-3 surfaces and their

rnoduli space.

The following theorem characterizes complex tori.

Theorem 17.4. A minimal surface S is a complex torus if and only if b I = 4,

K =-0. Moreover, a surface with 1< -~0 (necessarily minimal) has q = 0, Z, and

thus b I = 0, 4, 3 according to whether S is a K3 surface, a complex torus, or a

primary Kodaira surface.

61

As to c l a s s 4), w e r e c a l l t ha t we h a v e a l r e a d y i m p l i c i t l y r e m a r k e d t h a t the

two s u b c l a s s e s a r e d i s t i n g u i s h e d by the v a l u e of P I Z b e i n g 1 o r ~ Z, we h a v e in

fact the following.

Proposition 17.5.

Kod(S) = -~ if and only if Pi2 = 0.

Kod(S) = 0 if and only if PI2 TM I.

Kod(S) = 1 if and only if imi2 a Z and K Z = 0 for the minimal

model of S.

In p a r t i c u l a r , f o r the s u b c l a s s of c l a s s 4), w h e r e Kod = 0, we h a v e a r a t h e r

nice characterization.

Theorem 17.6. A surface S is h~perelliptic if and only if PlZ = I, b I = 2. A

surface with PlZ = I, b I = 0 is a K3 surface if pg 1 and an Enriques surface if

pg = 0 (then PZ = I).

For the subclass of class 4) with Kod = I, we see from proposition 17.5

that a characterization of a minimal model S is

(17.7) K x = 0, PlZ > Z, b I even

but, if we are given a non minimal surface, then the conditions b I even, and

[IZK I yielc~ing a rational map with image a curve are easier to check. As a

matter of fact, to detect the exceptional curves of the I kind on a surface with

Kod a 0 (i.e. PlZ ~ 0), the standard way is to look at the smallest m such that

P ~ 0, and then to look at the fixed part of Imlil, to check which components of m

this divisor are exceptional.

:Shrfaces of general type being taken into account by theorem 17. Z.v), we

notice that the surfaces in classes 6) and 7) are the ones with b I odd and, in partic-

ular, they cannot be Ka'hlerian. From surface classification and results of Kodaira,

Miyaoka, Todorov and Siu follows also the remarkable

T h e o r e m 17 .8 . A c o m p l e x s u r f a c e w i t h b 1 e v e n is a d e f o r m a t i o n of an a l g e b r a i c

s u r f a c e and is K ~ l e r i a n .

The two c l a s s e s 6) and 7) a r e d i s t i n g u i s h e d by the v a l u e of P 1 2 ' w h i c h is ~0

f o r c l a s s 6), and 0 f o r c l a s s 7) ( r e m a r k , t h o u g h , t h a t K o d a i r a ' s c l a s s VII is d i f f e r -

e n t f r o m o u r c l a s s 7), b e i n g d e f i n e d by t h e c o n d i t i o n b 1 = 1, and t h e r e b y i n c l u d i n g

62

also secondary Kodaira surfaces and some canonically elliptic surfaces). The sub-

class of Kodaira surfaces is again characterized by

(17.9) PI2 = i, b I odd

and we defer the reader to [IKo 4] for a very detailed description of these surfaces.

For lack of time, we don't attempt to describe the known examples and the

classification results for surfaces in class 7), referring to [Nak i] for a nice and

updated survey.

Let's just notice that, when b I = l, then, by a result of Kodaira on elliptic

surfaces, we have pg 0, and the intersection form Q is semi-negative definite

by 15. 18. More precisely, by Noether's formula 15. 15, since X = 0, b 2 = b z = e

Z KZ = -K Hence, as in the case of ruled surfaces with q ~ 2, < 0 as soon as the

Betti number b 2 is ~ 0. On the other hand, if S is elliptic, by Kodaira's canoni-

cal divisor formula (cf. [B-P-V], pgs. 161-164), a rnultipie mK of I~ is linearly

equivalent to a multiple rF of a fibre F of the elliptic fibration, hence in particu- 2

lar K = 0 (the above canonical divisor formula shows also that a surface S ad-

mits more than one elliptic fibration only Lf it is not canonically elliptic and in fact

only if Nod = 0, since either r = 0 or I~ determines the elliptic fibration).

We have thus the following

Theorem 17.10. A mini~aal surface S has K Z < 0 if and only if either

with q ~Z (b I odd) or S has b I = I, b 2 > 0.

f o l l o w s

S is ruled

K 2 Since X = + e, by Gastelnuovo's theorem 17.2.vi) and the above one

Theorem 17. ii. A surface S has X < 0 if and only if S is ruled with q m 2.

63

LECTURE SEVEN: SURFACES OF GENERAL TYPE AND THEIR MODULI

§18. Surfaces of general type, their invariants and their geometry

Let's observe that Noether's theo-em 15. 15 and the index theorem 15. 16

ensure that the analytically defined invariants K 2 and %< are determined by e

and r , therefore I< 2 and X are topological invariants and the advantage of deal-

ing with them stems from the fact that, unlike K Z and e, they don't have to satisfy

any congruence relation° Also, by theorem 17o 2, we have K2 , i -> 1 for a mini-

mal surface S of general type (these are called Castelnuovo's inequalities), and

two more inequalities are satisfied.

(18. i) I< 2 - 4 ~ ZX - 6 (Noether's inequality) >_ Zpg

2 K g 9X ( B o g o m o l o v - M i y a o k a - Y a u ' s i n e q u a l i t y )

In f ac t , by c l a s s i f i c a t i o n , the i n e q u a l i t y is t r u e fo r a l l s u r f a c e s , e x c e p t fo r r u l e d

s u r f a c e s of i r r e g u l a r i t y q ~ 2, w h i c h have K Z = 8 ( l - q ) , X = 1 - q. S. To Y a u [ Y a ]

p r o v e d i n d e e d a m u c h s t r o n g e r t h e o r e m , in p a r t i c u I a r it f o l l o w s f r o m h i s r e s u l t s

the f o l l o w i n g

Theorem 18.2. If a surface of general type S has K z = 9X, then the universal

cover S of S is biholomorphic to the unit ball in ~Z .

An easily proven but nice corollary concerns the surfaces S for which the inter-

section form Q is positive definite: in fact, if b 2 b + KZ = = r , since ~ 9$<,

4K 2 3(K2+ e), K 2

= 3(12X) = i.e., - 2e ~ e, which is inturn equivalent to 3T ~ e =

Z +b 2 - 2b I. Then 2bz ~ 2(i - bl) and thus b2 = i, bl = 0, pg = 0 (b+=l =I +Zpg)

hence X = I, <2 = 12 - 3 = 9o

Corollary 18.3. The only complex surfaces for which the intersection form Q is

positive definite have b 2 1 and KZ 9, X = h they are either ~72 = = or a surface

of general type with the unit ball as universal cover.

By a result of Kodaira, the plurigenera are completely determined by the

invariants K z , X ; we have in fact

Theorem 18.4. If S is a minimal surface of general type, and m > 2,

P = X + (i/Z) KZm (m- I). m

64

A s a c o n s e q u e n c e of t h e t h e o r y of p t u r i c a n o n i c a l m a p p i n g s , t h a t we a r e g o i n g

now to explain, there follows the result that surfaces with given invariants KZ , X

helong to a finite number of deformation types.

Let's go back to the canonical ring f~ (S), defined in 15. Ig; 18.4 tells us that

its Hilbert polynomial is determined by K 2 , X, and clearly two minimal surfaces

of general type S, S' are ison-lorphic if and only if R(S) and f£(S') are isomorphic

graded rings° Before mentioning directly how to recover S from ¢£(S), let's re-

mark that, ~(S) being finitely generated, there exists an m such that every func-

tion in ~(S) can be written as a fraction whose numerator and denominator are th

sections of H°(@s(mK)). In other terms, there exists an m such that the m

is birational onto its image G Unfortunately, unlike pluricanonical map <0 m m

the case of curves, one cannot expect ~ to be an embedding: in fact, though m

K- C e 0 for each irreducible curve on S (17.2. ii)), there can be a finite number

of curves C ~- pl with K, G = 0 (E = -Z), and K is ample iff these curves do not

exist on S. Otherwise, since ;%(S) is finitely generated, one can take iK =

Proj ~,(S) (its points correspond to maximal homogeneous ideals in ~(S)), and

there exists a holomorphic map I~: S -~ X satisfying the following properties

(18.5) i) X is a normal surface

ii) if i: ix] ° -~ X is the inclusion morphism of the non-singular part of

X, then the sheaf of Zariski differentials w X = i,(fl~o) is invertible X

and II. (tu X) : @S(KS) (i.e., X has only Rational Double Points,

RoD. P. 's, as singularities).

iii) every pluricanonical map ¢Q : S-~ E factors through I] and m m

: X - ~ E m rrl "

Definition 18o 6. X = Proj (¢%(S)) is called the canonical model of S.

We defer the reader to [Ca 3] for a survey of recent results on pluricanonical

maps of surfaces of general type, and we content ourself with stating a by now

classical result of Bomblerio

Theorem 18.7. If m -> 5, ~ : X -~ E is an isomorphism° m m

65

§19. P l u r i c a n o n i c a l i m a g e s a n d G i e s e k e r ' s m o d u l i v a r i e t y

R e c a l l t h a t a p r o j e c t i v e v a r i e t y E c p N i s s a i d to b e p r o j e c t i v e l y n o r m a l i f

t h e r e s t r i c t i o n h o m o m o r p h i s m s

H°(@pN(k)) -~ H°(@E(k))

are surjective for each integer k >- 0. In the case where E = E m

canonical image of a surface S of general type we have (cf. [Ci 2]).

Theorem 19. I. If m >_ 8, ~ : X-~ E m m

is the pluri-

normal surface.

i s an isomorphism onto a projectively

We are going now to discuss very loosely the main line of ideas which lead to

Gieseker's theorem about the existence of moduli spaces of surfaces of general

type. First of all, let's recall Mumford's definition (cf. [Mu 2]).

Definition 19. 2. A variety ~K2 is said to be a coarse moduli space for sur-

,X 2 faces of general type S with given invariants K , X if there exists a bijection X

between ~ 2 and the set of isomorphism classes iS] of minimal surfaces as

K,X above, satisfying the following property: for each deformation g J~ 13 of such

surfaces, there is given a unique morphism ~/ : 13 ~ ~ 2 such that ~b(b) = K,X

k-l([Sb]), and the correspondence (~ p B) -~ ~ is compatible with pull-backs

(i.e., to the family f g -~ B° corresponds d 2 o f = ~ t: 131 _~ ~ ) . K2,~<

Theorem 19.3 (Gieseker). There exists a coarse moduli space ~ 2 which is

K,X a quasi-projective variety. Moreover, two surfaces S, S s correspond to points in

the same connected component of ~ 2 if and only if S is a deformation of St

K,X th

The key point consists in taking all the m canonical images E of our 2 m

surfaces (with K , "X fixed): they are, if m -> 8, projectively normal surfaces of

fixed degree (= m2K 2) in a fixed projective space ~ of dimension P - 1 =

~4- i + m(m-l)/2 K2 . m

Now, E and E s are projectively equivalent if and only if the correspond- m m

ing surfaces S, S t are isomorphic, and one has to construct a quotient by the group

PGL(P ) of a variety .~ parametrizing our surfaces E . To this purpose, one m m

has first to use the Hilbert scheme (cf.[G~2], [Mu 3], [Ser Z]) technique: since

these notes are meant to be elementary, let's indicate the main idea.

66

Since E = E is projectively normal, if we denote by J m

E in ~ , the s p a c e H°(~E(n)} has f i x e d d i m e n s i o n e q u a l to

n + Pro) _ p P nm m

the ideal sheaf of Z

Now, there exists an integer n, depending only on the Hilbert polynon~fial P(n) of

a variety (here P(n) = X + IcZ/Z nm(nm- i), hence P depends only upon KZ, x,m),

such that the ideal sheaf ~ of Z is generated by the subspaee V Z= H°(~9 z(n)) of

the fixed vector space W = H°(@ip(n)). Let r be the dimension of V , t the di- 2

mension of W: then r = t - P(n), and r,t depend only upon K , X, m.

Definition 19.4. The n th Hilbert point of E is the point [P(A r V E) E [P(A r W),

belonging to a Grassman manifold G(r,W) of dimension r- P(n)o

The Hilbert scheme of subschemes of • with Hilbert polynomial ~(n) is the

closed subscheme 5C of the Grassmannmanifold G(r,W) c [p(ArW) corresponding

to the set of r-dimensional subspaces V of W such that the ideal sheaf ~= V@ P

generated by V defines a subscheme E(V) with Hilbert polynomial P(n). I£ is

the basis of a universal family, Joe., there is a subscheme Z of %~× ~ such that

the fibre of Z over V ( K is just the subscheme E(V). This %< is too big, and

first of all one has to take the open set .'< c K o

(19.5) K o

: [ V E W I V@p defines a connected surface ~(V) with only

R.D.P.'s as singularities} ,

Over .~ lies the restriction Z of the universal family Z, and inside ~ lies o o o

the closed subscheme

(19.6) ~ = { V i E = E(V) has WE(-1) --~ @Z ] o

The restriction Z of the universal family enjoys the following

19,7. Universal property of Z c ~ X rP: for each family p: ~ -~ B of minimal

surfaces of general type, with given invariants K 2, X, and for each

choice of P independent sections of the locally free sheaf ® I~ m

P~:<(~SIB )' t h e r e does e x i s t a unique p a i r of m o r p h i s m s

such that the following diagram commutes

67

and such that

pl f i 13 >

th gives fibrewise the m -canonical map

m: Sb -~ ~ "

Now, all the surfaces of general type with those given invariants II Z, X appear as

(minimal resolutions of) fibres of the family Z -~ .~ of canonical models; since the

base ~ is quasi-projective, hence it has a finite number of components, and, by

a result of G. Tjurina [Tju] , the canonical models X, X' of two surfaces of

general type S, S' are a deformation of each other if and only if S, S t are defor-

mations of each other, it is thus proven (cf. 19.3).

( t 9 . 8 ) The surfaces of general type with given I<2 Z, X belong £o a finite number

of deformation classes. In particular there is only a finite number of

dlffeomorphism type s.

It is also clear that the group

(19 .9 ) G'= PSL(P ) acts on ~, and there is a bijection between the set of m

orbits of G and the set of isomorphism classes of surfaces of 2

g e n e r a l type w i t h i n v a r i a n t s K , X.

Thus, the problem of the existence of the coarse moduli space T~ Z is reduced

K,X to the existence of a categorical quotient (cf [Mu 2], chap. I) for the action of G

on ~. This is a problem in the realm of the so called Geometric Invariant

Theory: in general (cf. [Mu 2], Appendix) such quotients always exist as alge-

braic (or Moishezon) spaces (i.e., as complex spaces bimeromorphic to algebraic

varieties). In fact, we recall again, what Gieseker proves is that

(19. I0) The categorical quotient ~/G ~ exists as a quasi-projective variety.

The idea is as follows, the Hilbert point belongs to

P [P(A r W) = ~P(A r (symn(~E m)))

and one wants to show that the inv'ariant homogeneous functions of degree ~ , i.e.

the sections in

68

Pro) G ' A = H ° ( p ( A r ( s y m n ( c ) ) , ~ ( ~ ) ) ,

f o r m , n, I s u f f i c i e n t l y b ig s e p a r a t e the o r b i t s , so t ha t ~/G' s i t s i n s i d e a p r o j e c t ~

t i v e v a r i e t y . We h a v e h e r e the t y p i c a l s i t u a t i o n of G e o m e t r i c I n v a r i a n t T h e o r y : a

vector s p a c e (SY mn (cpm U= h r )),

and an action of SL(P ) = G Then m

(19.11) A point u E U is

I) unstable if Gu 9 0

2) semistable if Gu ~ 0

3) stable if it is semistable and the

stabilizer of u is finite.

and then the G invariant polynomials define a morphism d~ from ~°(u)SS =

~ semi- stable points} to a projective variety Y in such a way that the restriction

of ~ to the stable points ~(U) s separates the orbits (in fact, the stable points

have closed orbits, and two closed orbits are separated by some G invariant

polynomials.

Remark 19.12. In our case the condition that the stabilizer of u be finite, when

u corresponds to the Hilbert point of a pluricanonical image ~ follows from a m

general result of Matsumura [Mat] . In fact, {g I g(E ) = Z } = Aut(Z) is a m m

linear algebraic group which, if not finite, would have a non-trivial Caftan sub-

group, which is a rational variety: but then E would be uniruled (X, with

dim X = n is said to be uniruled if there exist Y, with dim Y = n-l, and a domi-

nant rational map of Y × ~ °l into X), and in particular all the plurigenera of

would vanish.

The r e a l l y d i f f i c u l t po in t is to p r o v e tha t t h e s e H i l b e r t p o i n t s a r e s e m i -

s t a b l e , and t h i s is done wi th h a r d c o m b i n a t o r i a l e s t i m a t e s u s i n g the H i l b e r t -

M u m f o r d s t a b i l i t y c r i t e r i o n .

1 9 . 1 3 .

t E C

u is s e m i - s t a b l e if and only if fo r e a c h l - p a r a m e t e r s u b g r o u p of G ,

-~ g(t), w h e r e

t o 0

g(t) = A - . a ~ A -1 (with Z a. = O) 0 "t p / I

one has lira g(t)(u) ~ O.

69

From the fact that ~/G = ~9 2 is a coarse moduli space (cf. 19.2) it K,X

follows easily the following

Corollary 19. 14. Let S be a minimal surface of general type with invariants

I<12, X: then, if B is the base of the l~uranishi family of S, then, locally around

the point iS] corresponding to the isomorphism class of S, ~ 2 is analytically

isomorphic to the quotient of B by the finite group Aut(S).

§ 20. The number of moduli M(S) of a surface

Definition 20. I. For a surface of general type S , we define M(S), the number of

moduli of S, £o be equal to the dimension of the base ]5 of its IKuranishi family,

(i. e. the maximum of the dimensions of the irreducible components of B.

Remark 20. 2. By 19. 14, M(S) is the dimension of 9~ 2

responding to S. Iq , 5(

at the point iS] cor-

Remark 20.3. More generally, Kodaira and Spencer ([K-S], Chap. V, §ii) define

the number of modull of a complex manifold 34 to be the maximal dimension of the

base T of an effectively parametrized family Z -~ T of deformations of X, i.e.

such that the Kodaira-Spencer map O t is injectlve for each t .

It was conjectured by Noether that M(S) would be 10X - 2K 2 , whereas

Enriques realized that I0~( - 2K 2 was only a lower bound for M(S). In fact, by

the Hirzebruch-Riemann-Roch formula (cf. [Hi]), -(I0 X - Zl<[ 2) equals the Euler-

Poincare characteristic of ®, i.e. h°(~) - hl(®) + h2(@). In general, H°(~ X) is

the Lie algebra of a real Lie group of biholomorphisms of X: since (cf. 19. IZ)

Aut(S) is finite, H°(®S) = 0, and, as we saw in 3.3.

(20.4) 10X- 2K 2 = hl(®)- h2(®) ~ dimB = M(S).

On the other hand, M(S) ~ hl(®s) , and we indeed conjecture that for the

general surface S in each component of the moduli space ~ 2 equality holds,

and K is ample K , provided q(S) = 0--V~or reasons stemming from [Ca 4] and [Ca 5]), so that

should be aireduced variety. In any case, finding the dimension hl((@) is by no

means easier {and in some cases even more difficult) than to compute M(S),

h2(®S ) i ® f2), hence therefore we just observe that = h°(QS

70

(20.5) M(S) ~ hl(®) = I0 X - 2K2 + h°(Ql @ 2) ,

and one can give an upper bound for M(S) by giving an upper bound for h°(Ql® f2Z)

in terms of X, K2, q. It is clear that, doing so, one does not obtain the best esti-

mate, because one is giving an upper bound for the dimension of the Zariski tangent

space of each point of the base B of the IKuranishi family, and not simply an npper

found for dim B, as we already noticed. 1

C'/c z) = h°(Q~l(K)) is to use the existence of a A way to bound h°(flS ® o

K Z smooth curve C in IZKI as soon as ~ 5, or pg >- I, except possibly if

I< Z = 3, 4 (this follows from recent results of Francia [Fra] and Reider [Rei 2]).

In fact then, by the exact sequences

0 -~ ~I(_ II) -~ ~l(I<) -~ flIs(K)® ~C -~0

0 -~ ~C(-K) -~ f}sI(K) ® ~c ~ ~C (4K) ~ 0

since f%S l(-I<) ~ ®S ' we get

h°(f~Is(K)) ~ h°(@C(4Ii)) = 5K2 .

Otherwise, one looks for a smooth curve in linK l, with m=3, and the result is

(with an improvement only of the constant in [Ca 6], Thm. B)o

Theorem 20.6. We have the inequalities

10X- 2K 2 ~ M(S) ~ I0~< + 3K2+ 18 .

If II< I contains a smooth curve, then M(S) ~ 10% + q + I .

In the case when the surface S has q ~ 0, since q = h°(QS 1 ) , surely

s s h ° ( ~ ® 0S) ¢ 0, s ince there is a b i l i n e a r map H°(~ ) × H°(f~ )-'~H°(fl ® 0 S)

which is n o n - d e g e n e r a t e in each factor , n e v e r t h e l e s s the ex is tence of ho lomorphic

1 - fo rms can be used for our d e s i r e d bound.

For i r r e g u l a r (q ~ 0) su r faces , a powerfuI tool is given by the ana ly s i s of

the Alhanese map

H ° f/IV H 6: S ~ A =AIb(S) = ( S ) / I(S, 2)

P /-

such that c~ (p) = ~ / (where Po is a fixed point and the linear functional on

Po

HO(f~ i) is clearly defined only up to Jy , for y E HI(S, ~))° The condition that

71

s h o u l d h a v e d i f f e r e n t i a l of m a x i m a l r a n k = Z e x c e p t t h a t a t a f in i t e s e t of p o i n t s

c a n a l s o be p h r a s e d a s

t h e r e do e x i s t ~31' 132 E HO(Q;)t s u c h t h a t C = div(~] 1 A (Z0.7) ~2 )

is a reduced and irreducible curve (in II~ I !) •

C a s t e l n u o v o [ C a s ] t r i e d to p r o v e the i n e q u a l i t y M <- pg + 2q u n d e r the a s s u m p t i o n

t h a t c~ (S) wou ld be a s u r f a c e , i° e. , a s s u m i n g tha t the d i f f e r e n t i a l of c~ wou ld h a v e

r a n k Z o u t s i d e of a c u r v e : we s h o w e d in [Ca 6] t h a t t h e r e a r e i n f i n i t e l y m a n y

f a m i l i e s of s u r f a c e s S fo r w h i c h co(S) is a s u r f a c e , and M >- 4pg + O(pg). In

f a c t , we a l s o p r o v e d t h a t (20 .7) i m p l i e s M ~ pg + 3 q - 3, and c o n j e c t u r e d t h a t

(20 .7) wou ld i m p l y the C a s t e l n u o v o i n e q u a l i t y . We s h a l l now give R e i d e r ' s s i m p l e

p r o o f [ R e i 1] of t h i s i n e q u a l i t y .

T h e o r e m 20 .8 ( R e i d e r ) . If S s a t i s f i e s 2 0 . 7 , t h e n M g pg + 2q o

Proof. Since C is irreducible, ~] and 13 don't vanish simultaneously on a 1 2

curve, hence we can assume ~q = ~] 1 to have only isolated zeros. Let Z be the

0-dimensional scheme of zeros of ~3: we have then the l<oszul complex

(20.9) 0 -~ a S ~] * I A~q 2 0S ~ 3Z OS -~ 0

where =9 Z is the ideal sheaf of Z. Tensoring with @S(Ii) , since hl(®) =

hI(Qs(K)), we get

hl(®s ) Z (20.10) -< q +hl(jz g/S) - 1 ,

The basic fact is that the ideal sheaf of C is contained in ~9 Z'

exact sequence

hence we have an

(Z0. II) 0 ~ @s(-C) ~ JZ ~ ~ ~ 0 ,

where $ is a torsion free, rank 1 sheaf on Co Tensoring (20. ii) with @S(211)

yields

(20. 12) hl(az(Z S)) <- q +hl(~(Z S ))- 1 .

Since @C(2K S) is the dualizing sheaf of C, by Grothendieck duality

hl(~ (2KS)) : dim Hom(~ (KS), %C(K)) °

72

On the other hand, there is a bilinear map

H°(Z(K)) X Hom($(K), @c(K)) -~ H°(@c(K))

which is non-degenerate in each factor; since these are complex vector spaces, a

well known application of the Segre product (projectived tensor product), gives the

inequality

hI($(ZKS) ) < h°(@C(K)) - h°($(K)) + 1

pg + q- 1 -h°(~(l<)) + I o

To give a lower bound for h°(~ (I<)), we tensor (Z0. II) with @S(I<), to get

h°($ ~<)) ~ h°( ~9 Z(I<)) - I >- q - g by the sequence (Z0.9). It suffices to put these

inequalities together. Q.E.D.

W e d e f e r t h e r e a d e r to [ R e i 1] f o r o t h e r r e s u l t s o f t h i s t y p e , a n d w e n o t e i n

f a c t t h a t R e i d e r s i m p l y u s e s t h e e x i s t e n c e o f a l - f o r m ~ w i t h i s o l a t e d z e r o s , a n d

t h e e x i s t e n c e o f a r e d u c e d i r r e d u c i b l e c u r v e C i n I K t s u c h t h a t @ s ( - C ) c 3 Z ,

so that the method can be generalized taking C E link i with such a property. It

would be interesting to give an upper bound for M(S) in the case of a surface S

fibred over a curve B of genus m Io

73

LECTURE EIGHT: BIHYPERELLIPTIC SURFACES AND PROPERTIES OF THE MODULI SPACES

§21. Moduli spaces of surfaces of general type and their properties 2

Let S be a minimal surface of general type with inv-ariants K , X and

consider the Gieseker moduli space ~ 2 , which has a finite number of con- K,X

nected components° We can define two moduli spaces ~top and ~diff which are

c o n t a i n e d in ~ll , and are indeed a union of connected components of ~ , 2, X K2, %

as follows (~diff c ~top).

Definition El. I. ~t°P(s) (resp.: ~diff(s)) is {[S'] E ,~ 2 I there exists

K ,X an orientation preserving homeomorphism (respo: diffeomorphism) between

S and S'].

We n o t e (cf . § 1 4 . 17) t h a t i f Q i s e v e n a n d K Z > - 9, t h e n e v e r y c o m p l e x s t r u c -

t u r e on the topological manifold underlying S corresponds to a minimal surface S'

of general type homeomorphic to S. Thus ~diff , e. go, is then a coarse moduli

space for all the (integrable almost) complex structures on the differentiable mani-

fold underlying S.

Remark 21o 2. Since the Stiefel-Whitney class w 2 (cf° 14) is the mod 2 reduction

of cl(K ) E HZ(s, ~), we see that the intersection form Q is even if and only if

Cl(K ) E 2H2(S, 7/). Therefore, Freedman's theorem 14°3 has as a corollary that

two simply connected complex surfaces S], S Z are homeomorphic if and only if

they have the same invariants I <12, X, and for both of them the same answer holds

true to the question: does K. E 2HI(s ,2~) or does it not? I I

We note that in complex dimension n = I, the notion of homeomorphic and

diffeomorphic are the same, and that the moduli space 7~ of curves of genus g g

is connected, also irreducible, and of pure dimension 3g- 3 (more of its proper-

ties a r e k n o w n , cf° [ H - M ] , a n d H a r e r ' s n o t e s in t h i s v o l u m e ) .

In the n e x t p a r a g r a p h s , we s h a l l c o n s i d e r s o m e f a m i l i e s of s u r f a c e s , s o m e -

h o w a g e n e r a l i z a t i o n of h y p e r e l l i p t i c c u r v e s , b y w h i c h we s h a l l s e e t h a t ~ t o p d o e s

n o t s h a r e a n y of t h e s e t h r e e good p r o p e r t i e s w i t h ~ We h a v e in f a c t t h e f o l l o w - g

ing (of. [Ca 6 ] , [ C a 8 ] ) .

T h e o r e m 2 1 . 3 . F o r e a c h n a t u r a l n u m b e r k, t h e r e e x i s t s a m i n i m a l s u r f a c e of

g e n e r a l t y p e S s u c h t h a t ~ t ° P ( s ) h a s a t l e a s t k i r r e d u c i b l e c o m p o n e n t s

74

Y Y with 1 ..... k

i) dim(Y i) ~ dirn(Y.) for i ~ j J

ii) Y. and Y lie, for i ! j, in different connected components of ~t°P(s)° i j

It is not clear at the time being how smaller than ~top is really ~diff :

Donaldson [Do 2] has shown the existence of two homeomorph{c, but not diffeo-

morphic, complex surfaces, so we should expect to have ~diff ~ ~top in general,

and ~diff could still have some nice properties.

§ 22. Bid0uble covers and their deformations

Definition 22. I. A smooth bidouble cover (it is a fourfold cover) is a Galois finite

cover, ~: S -~ X with group 2/2 @ 2/2, between smooth varieties.

Example 22.2. Let co: ~2 ~2 be the morphisrn definedby (x0, xl,x Z) -~

(x02'Xl 2'x2) = (z0'zl'z2)° The group (~/Z) 2 acts with the three covering involu-

tions ~0' C;l' C52 such that c5. (x.) = x. if i ~ j, ~.(x.) = -x.. The fixed locus l j j i i l

for ~. is the line ix. = 0] = R. plus the point R. • i< k , if (i,j,k) is a permuta- 1 1 i J

tion of (i, 2,3), and the branch locus B = ]B 1 + 22+ B 3 consists of the 3 coordinate

lines, respective images of RI, R2, 1<3 ° This example, though easy, gives all

the ingredients of the geometric picture in the general situation.

As in the example one defines (cf. [Ca 6]) ~0' ~i' ~2 to be the three non-

trivial elements of the group, and denotes by R. the divisorial part of the fix 1

locus of ~. , by B. the set theoretic image of Ft . 1 1 1

If z = 0 is a section of @X(Bi) with div(z.) = B , we see from the example 1 1 1

above that it is not possible to take directly only the square root x. of z. , but I 1

that it is possible to take the square roots w.l of Z0ZlZ2/Z i (these three double

covers correspond to the three distinct subgroups of order 2 of (~/2) 2, giving

each a factorization of ~ as a composition of two double covers). In general, thus,

there are divisors L 0, L I, L Z on X such that 2L.~ ~ B0+ BI +22 - Bi '

B k + L k m L 0 + L 1 + L 2 - L k , and S is the smooth surface, in the rank 3 bundle

on X which is the direct sum of the three line bundles corresponding to the L.'s, i

defined by the equations l' I Z0ZlZz/Zi

(22.3)

ZkW k = W0WlWz/W k

75

There is a natural way of deforming these equations, and, computing % and the

characteristic map of the family, one can check whether the characteristic system

of the morphism M is complete.

We refer to [Ca 6] for more details, and observe that equations 22°3 take a

much simpler form, and the natural way of deforming is easier to see if one

assumes one involution, say c~ 0 , to have only isolated fixed points, i.e. , if one

assumes z 0 = i.

D e f i n i t i o n 2 2 . 4 . A s i m p l e b i d o u b l e c o v e r i s a s m o o t h b l d o u b l e c o v e r s u c h t h a t o n e

o f t h e t h r e e c o v e r i n g i n v o l u t i o n s h a s a f i x e d s e t o f c o d i m e n s i o n a t l e a s t Z.

The equations 22.3 simplify then (set x I

(22.5)

2 x I =z I

x 2 = z 2 ,

= w~ Z , x 2 = w I) to

and a natural way to deforming them is to set

(22.6)

Z x I = z I + blX 2

2 x 2 = z 2 + b2x I

for b. a section of @x(Bi i - Li). In general, applying ~P.

0 -~ $S -~ M ® X -~ N -~ 0 ,

one obtains

to the exact sequence

(22.7) 0 -~ cP,® s -~ ®X ® (@X 9 @x(-LI ) ~ %x(-L2 ) G@x(-L3))

2

G) (0BI(Bi). ~ @B. (B'I - L.))I -~ 0 i=O i

Now, the parameter space for the natural deformations of cp

2

@ H°(@x(Bi) e ~x(Bi - Li) ) i=0

a n d o n e o b t a i n s

i s the vector space

Theorem 22.8. The characteristic system of the map cp is complete if

H°(@x(Bi)) goes onto H°(@ B (Bi)), and the same holds for I

76

H°(~x(B i - Li)) -~ H°(@B.(B i - Li)) I

If furthermore HI(® X) = HI(®IK(-Li)) = 0, then the Kuranishi family of S is

s m o o t h .

A c t u a l l y t h e h y p o t h e s e s in t he a b o v e t h e o r e m a r e r a t h e r r e s t r i c t i v e , a n d

no t s t r i c t l y n e e d e d , a n y h o w t h e y a r e s u f f i c i e n t f o r o u r a p p l i c a t i o n (cf . § 23). A l s o ,

a s i m i l a r r e s u l t s h o u l d h o l d t r u e m o r e g e n e r a l l y f o r s m o o t h A b e l i a n c o v e r s .

§ 23. Bihyperelliptic surfaces

Hyperelliptic curves are double covers of pl , and if we multiply all the

previous terms by two we get

Definition 23. i. A bihyperelliptic surface is a smooth bidouble cover of ~?I x [pl .

In the following, we shall limit ourselves to consider simple bihyperelliptic

surfaces. These are determined by the two branch curves B I, B Z of respective

bidegrees (2n, 2m), (Za, Zb). One can allow also the two curves B I, B 2 £o

acquire singularities, but in such a way that the bidouble cover defined by equa-

tions (22.5) have only R.D.P. 's as singularities: we shall call the resulting sur-

faces admissible.

(z3.2) D e n o t e b y ~ ( a , b ) ( n , m ) t h e s u b s e t of t he m o d u l i s p a c e

c o r r e s p o n d i n g to n a t u r a l d e f o r m a t i o n s of s i m p l e b i h y p e r -

e l l i p t i c s u r f a c e s w i t h b r a n c h l o c i of b i d e g r e e s (Za, 2b),

(2n, 2m) . D e n o t e f u r t h e r by ~ ( a , b ) ( n , m ) t he s u b s e t c o r r e -

s p o n d i n g to a d m i s s i b l e s u r f a c e s ( t h e s e a r e t he s u r f a c e s

w h o s e c a n o n i c a l m o d e l s a r e d e f i n e d b y e q u a t i o n s ( 2 2 . 6 ) ,

and occur precisely when those equations give surfaces

w i t h a t m o s t R. D. P . ' s a s s i n g u l a r i t i e s ) .

A n e a s y a p p l i c a t i o n of t h e o r e m ZZ. 8 s h o w s

T h e o r e m Z 3 . 3 . ~ (a, b ) (n , m ) i s a Z a r l s k i o p e n i r r e d u c i b l e s u b s e t of t h e m o d u l i

space. In particular, the closure ~ (a, b)(n, m) is irreducible (and contains

(a, b)(n, m) ) •

77

Remark 23.4. i) Clearly

(a, b)(n, m) = ~(b, a)(m, n) = ?~(n, m)(a, b) = ~(m, n)(b, a)

apart from these trivial equalities, all the ~(a,b)(n,m) 's can be proven to be dif-

ferent (by the inflectionary behaviour of the canonical map of the general surface

in the family), and hence they are disjoint by theorem 23.3.

ii) Also, since R D1 X ~ D1 = iF 0 is a deformation of ~2n ' one can also show

(cf. [Ca I] ) that, enlarging the set ?~(a,b)(n,n~) to the smooth bidouble covers of

~2n ' and doing the same for the admissible covers, a result similar to 23.3

holds true.

iii) If a > 2n, m > 25 , it follows easily from equations (22.6) that all the ^

surfaces in ~(a,b)(n,m) are admissible (simple)bihyperelliptic surfaces.

We can now sketch the main arguments for the proof of theorem 21.3.

(23. s) Bihyperelllptic surfaces are simply-connected, and their

invariants K 2 , X are expressed by quadratic polynomials

P, Q of (a,h,n,m)°

(23.6) Also dim ?~(a, b)(n, m)

( a ,h , n, m) .

is given by an easy function of

(z3.7)

(23.8)

Letting r(S) = maxim I Cl(K ) E mHZ(S,I)] , since for a family

8-~ B, @St(Kt) = @St ® WglB , we have that r is a locally

constant function on the moduli space. Moreover (cf. [Ca 7]

for the proof, using easy arguments of group cohomology),

if iS] E ~(a,b)(n,m)' then r(S) = G.C.D. (a+u-2, b+m-2).

One has to show (this was done by Bombieri in the appendix to

[Ca 6], that for each k, there exist k 4-£uples (a,b)(n,m) 2

giving the same values for I~ , N, and k distinct values for

both Iv[(~ and r(S), and one can further assume r(S) to take even

values. But when w 2 = 0, Q is even, the S's are simply con-

nected, therefore (cf. 21.2) one gets k distinct irreducible com-

ponents, of different dimensions, belonging to the same moduli

space ~t°P(s), and lying in k distinct connected components

of ~t°P(s). I conjecture the closures of the (a, b)(n, m) 's

(at least if a > 2n, m > Zb) to be themselves connected

78

components of the moduli space. The following has been proven

up to now ([Ca 7]), and it is an encouraging result, since one of

the most difficult problems is, in general, to describe deforma-

tions in the large of complex manifolds.

Theorem 23.9. If a > 2n, m> Zb the closure of ~(a,b)(n,m)

sible covers of some [Fzk , with

(. n) k d m a x a ~ - - 1 ' m - 1 "

consists of admis-

I n p a r t i c u l a r ~ ( a , b ) ( n , m ) is a closed subvariety of the moduli space if

a -> max(2n + I, b)

m ~- max(2n + I, m)o

Idea of proof: If a> 2n, m> 3b, then (cf. 23° 4. iii) all the surfaces in l

P ~(a,b)(n,m) are simple bihyperelliptic and, given a l-dimensional family ~ -~ T,

with _ _is t ] 6 ?~(a,b)(n,m) for t g to , one wants to conclude that St is still an O

admissible cover of some [Fzk . The key point is that (2/2) 2 acts birationally

on S, preserving the deformation morphism p' , but indeed it acts biregularly on

the family ~ ~ T of the canonical models of the previous family pt: 8 ~ T .

= f ( : ~ I 2 ) z , What we have to show is that, if Z % (then Z = IpIx for t ~ t ), t o

then Zto =~'2k " To achieve this goal it suffices to show that q: Z -~ T is a

smooth fibration, since every deformation of a minimal rational ruled surface with

Q even is again a surface of the same type. Now, the singularities that Zto

can have are quotients of R.D.P.'s by ~/2 or (2~/Z) 2, and can be explicitly

classified: but many of them can be shown not to occur since any smoothing of

these singularities would contribute, through the vanishing cohomology of the

Milnor fibre, a subspace of HZ(Zt ,2) of dimension >- 2 over which Q is nega- [pl

tire definite. This is a contradiction, since Z t ~I X , and other arguments

again by contradiction, eliminate the other remaining possibilities.

I should finally remark that to prove that the closure of ?~(a,b)(n, m) is open

in the moduli space, it would suffice (by the results of [Ca 6] and [Ca 7] ) to prove

also when the canonical model of S is singular (i.e., the bidouble cover is ad-

missible, but not smooth) that the base B of the IKuranishi family of S is locally

i r r e d u c i b l e .

79

Refe re nee s

[At]

JAr z]

[A-S]

[B-DF]

[B-P-V]

[Be I]

[Bo]

[Bo-Hu]

[Brc]

[B-W]

[Cas]

[ca l]

[Ca 2]

[ca 3]

[ca 4]

[Ca 5]

[Ca 6]

[ca v]

[Ca 8]

[ci l]

[ci z]

Artin, M. , "On the solutions of analytic equations," Invent. Math. , 5 (1968), 277-291, MR 38 #344. Art[n, M., "Algebraization of formal modul[, I, Global Analysis," Princeton Univ. Press, Princeton, IN.J., and Tokyo Univ. Press, Tokyo, 1969, 21-71, MR 41#5369. At[yah, M. Io ~ Singer, I. M° ,"The index of elliptic operators [II,"Ann. Math., 87 (1968), 546-604° Bagnera, G., De Franchis, M°~"Sopra le superficie algebriche che hanno le coordinate del punto generico esprimibili con funzioni meromorfe quadruplemente periodiche di 2 parametri;'Rend. Acc, de[ Lincei 16 (1907). Barth, W., Peters, Co, van de Ven, A., "Compact complex surfaces," Springer Ergebn[sse 4, (1984), Berlin-Heidelberg° Beauville, A., "Surfaces alg~briques complexes;' Ast~risque, 54 (1978). Bombieri, E., "Canonical models of surfaces of general type,"

FhJb]. Scient. I.H.E.S. 42 (1973) , 447-495. Bornbieri, E., Husemoller, D., "Classification and embeddings of surfaces," in Algebraic Geometry, Arcata 1974, A.IVi. S. Proc. Syrup. Pure iv[ath., Z9 (1975), 329-4Z0o Borcea, C., "Some remarks on deformations of Hopf manifolds, Rev. Roum. Math. Pures Appl., 26 (1981), lZ87-1Z94° B~rns, D., Wahl, J., "Local contributions to global deformations of surfaces," Inv. Math., Z6 (1974) ,67-88. Castelnuovo, G°, "Sul nurnero de[ rnoduli diuna superficie irregolare, I, II," Rend. Accad. Lincei, 7 (1949), 3-7, 8-11. Catanese~ F. , "Moduli of surfaces of general type, " in Algebraic Geometry- Open problems, Proc. Ravello 1982, Springer L.N.M. 99 v (1983), 90-11Z. C~tanese, F., "Superfieie complesse compatte, " Atti Convegno G.N.S.A.G.A. del C.N.R., Valetto, Torino (]986) Catanese, F. , "Canonical rings and "special" surfaces of general type,"

Proc. of A.M.S. Conf. in Algebraic Geometry, 1985. C~tanese, F., "Commutative algebra methods and equations of regular surfaces," in Alg. Geometry-Bucharest 198Z, Springer L.I%M. 1056 (1984), 68-111. Catanese, P., "On the period map of surfaces with K Z =X = 2," in Classification of Algebraic and Analytic Manifolds, Proc. Katata Syrup. 198Z, P.M. 33 Birkh~user (1983), 27-43. Catanese, F. , "On the moduli spaces of surfaces of general type, " J. Diff. Geomo, 19 (1984), 483-515. Catanese, F. , "Automorphisms of rational double points and moduli spaces of surfaces of general type," ~omp. Math. (31 119~7) 81-102. Catanese, F., "Connected components of moduli spaces," J. Diff. Geom. 24 (1986) 395-399.

Ciliberto, C., "~k~perficie al~ebriche complesse: idee e metodi della classificazione"in Atti del Convegno di Geometria Algebrica, Nervi 1984, Tecnoprint Bologna (1984), 39-157. Ciliberto, C. , "SUl grado de[ generatori dell' a nello canonico di una superficie di tipo generale, " Rend. Sere. Mat. Univ. e Polit. di Torino 41, 3 ( 1983 ) , 83 - 112 .

80

[Da]

[Do I]

[Do z]

[Dou 1]

[Don Z]

[Dou 3]

[Dou 4]

[Eh]

[En] [Fo]

[Fra]

[Fre]

[Gie]

[Gr I]

[Gr Z]

[Gr 3]

[G l]

[G-HI

[Oro i]

[Gro z]

[Hab ]

[H-M]

[Hi]

[gir]

[Hor o]

[Hor i]

Dabrowski, K°, "Moduli spaces for Hopf surfaces,'Math0 Ann., 259 (198Z), Z01-ZZ6o ]Donaldson, S. , "An application of gauge theory to four dimensional topology, ~' J. Diff. Geom., 18 (1983), 279-315. Donaldson, So, "Za topologie differentielle des surfaces complexes," C°R. Aead° Sc. Paris 301, I, 6 (1985), 317-3Z0. Douady, Ao, "Le probleme des modules pour les varifies analytiques complexes, S~m° Bourbaki (1964/65), expos~ Z77. Douady, A., "Flatness and privilege," in Topics in several complex variables, Monographie n. 17 de l'Ens. Math., Geneve (1968), 47-74. Douady, A. , "Obstruction prirnaire ~ la d~formation, S~minaire H. Cartan, 13CAnnee (1960/61), expos~ 4. Douady, Ao, "Le problgme des modules pour les sous-espaces analy- tiques compacts d'un espace analytique donn~, " Ann. Inst. Fourier (Grenoble) 16 (1966), fasc. l, 1-95, MR 34 #Z940. Ehresmann, C., '"'Sur les espaces fibres diffgrentiables," C. R. Acad. Sci. Paris, ZZ4(1947), 1611-161Z. Enriques, F., "Ze superficie algebriche," Zanichelli, Bologna (1949) Forster, O°, "Power series methods in deformation theory, " Proc. Symp.'Pure Matho, 30 (1977), A.M.S°, 199-217. Francia, P°, "The bicanonical map for surfaces of general type," to appear in Math° Ann. Freedman, Mo I-Io, "The topology of four dimensional manifolds, " J. Diff. Geom., 17 (198Z), 357-453. Gieseker, D,, "Global moduli for surfaces of general typeS' Inv° Math. 43 (1977), Z33-282. Grauert, H., "~ber die Deformation isolierter Singularit~ten analy- tischer Mengen," Invent. Math., 15 (1972), 171-18Z, MR 45 #2Z06. Grauert, H° , "~ber Modifikationen und exzeptionelle analytische Mengen," Math. Ann., 146 (196Z), 331-368. Grauert, H. , "Der Satz von iKuranishi f~r kornpakte kornplexe Ra{~me, Invent. Math., 25 (1974), i07-142, MR 49 #109Z0. Griffiths, Ph. A. , "Periods of integrals on algebraic manifolds, I. II, " Amer. J. Math., 90 (1968), 568-626, 805°865, MR 37 #5Z15, 38 #Z146. Griffiths, P., Harris, J., "Infinitesimal variation of Hodge structure II, " Comp. Math 50 (1983) ,Z07-Z65. Grothendieck, A., "Construction de l'espace de Teichm~ller, S~minaire If. Cartan, 13 c Annie (1960/61), l~pos~ 17. Grothendieck, A., "Techniques de construction II; Le th~or~me d t existence en th~orie formelle des modules, Sere. Bourbaki, exp. 195 Scre. Math. Paris (1960); Techniques de construction IV: Les schemas de I-lilbert, S~m. Bourbaki, exp. ZZI, Secr. Math. Paris (1961). Habegger, IW., "Une vartete de dimension 4 avec forme d'interseetion paire et signature -8," Comm. Math. Helv., 57 (1982), 22-24. Harris, J°, Mumford, D. , "On the Kodaira dimension of the moduli space of curves," Invent. Math., 67 (1982), 23-86. Hicks, N., "~4otes on differential geometry, " Van Nostrand, New York (1965). Hirzebruch, F., "Topological methods in algebraic geometry, Springer- Verlag, Grundlehren 131, 3 rd ed. (1966)o HorLkawa, F~,, "On deformations of holomorphic maps," in Manifolds - Tokyo 1973, lwanami-~oten (1974), 383-388. HIorikawa, E. , "On deformations of holomorphic maps I, " J. Math. Soc. Japan, 25 (1973), 372-396.

8]

[Hor Z]

[Hor s]

[Hor 4]

[H~,'r]

[Kas]

[Ko I]

[Ko z]

[Ko 3]

[Ko 4]

[K-M]

[K- N- s]

[z- s]

B<-s-z]

[Kn l]

[KuZ]

[~u 3]

[L-S]

[Man]

[Math t]

[Math 2]

[Mat]

[Mi Sta]

[Moi 1]

Horikawa, E. , "On deformations of holomorphic maps If, " J. Math. Soc. Japan 26, 647-667, III ibidem. Horikawa, E°, "On deformations of quintic surfaces, " inv. Math., 31 (1975), 43-85. Horikawa, E., "On the number of moduli of certain algebraic surfaces of general type," J. Fac. Sci. Univ. Tokyo (1974), 67-78. H6"rmander, Lo , "An introduction to complex analysis in several vari- ables, " IIed. , North Holland, Amsterdam (1973). lias, A. , "On obstructions to deformations of complex analytic sur- faces," Proc. Nat. Acad. Sci. U. SoA., 58 (1967), 40Z-404. Kodaira, K. , "On stability of compact submanifolds of complex mani- folds," Amer. J. Math., 85 (1963), 79-94, MR Z7 #3002. Kodaira, K., "A theorem of completeness for analytic systems of sur- faces, with ordinary singularities," Ann. Math., 74 (1961), 591-627. Kodaira, K., "On characteristic systems of families of surfaces with ordinary singularities in a projective space, " Amer. J. Math. , 87 (1965), ZZ7-256. Kodaira, m., "on the structure of compact complex analytic surfaces, I," Amer. J. Math., 86 (1964), 751-798. modaira, K. , Morrow, J., "Complex manifolds, " Holt, Rinehart and Winston, Ne~v York (1971). Kodaira, K., Nirenberg, L., Spencer, D. C., "On the existence of deformations of complex analytic structures," Ann. Math. (Z) 68, (1958), 450-459, MR 22 #301Z. I<odaira, K. , Spencer, D. C. , "On deformations of complex analytic structures, I, II, " Ann. Math. (2) 67 (1958), 328-466, MR 2Z #3009. Kodaira, K., Spencer, D. C., "A theorem of completeness for com- plex analytic fibre spaces," Acta Math., i00 (1958), 281-294. I<uranishi, M., "On the locally complete families of complex analytic structures, " Ann. Math, (Z), 75 (1962), 536-577, MR Z5 #4550. I<uranishi, M., "New proof for the existence of locally complete fami- lies of complex structures, Proc. Conf. Complex Analysis (Minneapolis 1964), Springer-Verlag,Berlin-Heidelberg New York, 1965, 142-154, MR 31 #768. Kuranishi, M. , "Deformations of compact complex manifolds, " Presses de l'Universite de Montreal (1969) L[chtenbaum, S., Schlessinger, M., "The cotangent complex of a morphism, " Trans. Amer. Math. Soc., 128 (1967), 41-70, MR Z5 #4550. Mandelbaum, R. , "Four dimensional topology: an introduction, " Bull. A.M.S. (1980), 1-159. Mather, J. N., "Stable map germs and algebraic geometry, " ManLfolds-Amsterdam, 1970, Springer L.N.M. 197 (1971), 176-193. Mather, J. N., "Generic projections," Annals of Math., 98 (1973), ZZ6-245. Matsumura, H. , "On algebraic groups of birational transformations, " Rend. Accad. Lincei Ser. 8, 34 (1963), 151-155. Milnor, J., Stasheff, J., "Characteristic classes," Ann. Math. Studies 76, Princeton Univ. Press, Princeton (1974). Moishezon, B. G., "On n-dimensional compact varieties with n alge- braically independent meromorphic functions, " Izv. Ak Nauk SSSR Ser. Mat. 30 (1966), 133-144, 345-386, 6ZI-656. English transl., Am. Math. Soc. Transl. (2), 63 (1967), 51-177.

82

[Moi 2]

[IViu 1 ]

[Mu z]

[Mu 3]

[Nak I]

[N~ Z]

[Noe l]

[Noe Z]

[Pa]

[Pe]

[po]

[Pou]

[Ram]

[Rei l]

[Rei 2]

[~]

[Ro]

[Sch]

[Sei i ]

[Sei 2 ]

[Sei 3 ]

[mr]

[~r 2] [m]

[~ 2]

Moishezon, B. G., "An algebraic analogue to compact complex spaces with a sufficiently large field of meromorphic functions. Izv. Akad. Nauk SSSR Ser, Mat., 33 (1969), 174-238, 323-367, 506-548. Mumford, D., "Further pathologies in algebraic geometry," Am. J.

Math. 84 (1962) ,642-648 . Mumford, D., "Geometric invariant theory, " 2nd edition (1982), with J. Fogarthy coauthor), Heidelberg, Springer (1965). Mumford, D., "Lectures on curves on an algebraic surface, " Annals of Math, Studies 59, Princeton Univ. Press, Princeton (1966). Nakamura, L, "VII 0 surfaces and a duality of cusp sirgularities, " in Classification of algebraic and analytic manifolds, Proc° Katata Syrup. 1982, P.M. 39, Birkh~user (1983), 333-378. Nakamura, L , "On surfaces of class VII o with curves, " Inv. Math,

78, 3 (1984), 393-443. Noether, M. , "Zur theorie des eindeutigen Entsprechens algebraischer GebHde," Math. Ann., 2 (1870), 293-316, 8 (1875), 495-533. Noether, M. ,"Anzahl der Moduln einer Classe algehraischer Fl~chen," Sb. Kgl. Preuss# Akad° Wiss. Math.-Nat. KI., Berlin (1888), 1Z3-127. Palamodov, V. P. , "Deformations of complex spaces, " Uspekhi Mat. Nauk. 31:3 (1976), 129-194. Transl. Russian Math, Surveys 31:3 (1976), 129-197. Persson, U., "An introduction to the geography of surfaces of general

type, " Proc. of A. M.S. Conf. in Algebraic Geometry, 1985. Popp, H., "Moduli theory and classification theory of algebraic varie- ties," Springer L.N.M. 620 (1977). Pourcin , G., "Deformation de slngular~tes isol~es, Ast~ri~qu¢, Soc. Math. France 16 (1974), ]61-173. Ramanujarn, C. P. , "Remarks on the Kodaira vanishing theorem, " J. Indian Math. Soc., 36 (1972), 41-51; supplement in J. Indian Math. Soc. 38 (1974), 121-124, Reider, I., "]Bounds on the number of moduli for irregular surfaces of

general type, " Manuser. Math. Reider, L, ,t Linear systems and vector bundles of rank 2 on surfaces" Annals of Math., to appear.

Rokhlin, V. A. , "New results in the theory of 4-dimensional manifolds, " Dokl. Akad. l~auk. SSSR, 84 (1952), 221-224o Rossi, H,, "Vector fields on analytic spaces," Annals of Math. 78, 3 (1963), 455-467. Schlessinger, M., "~-hlnetors of Artin rings," Trans. Amer. Math. Soc.; 130 (1968), ZO8-ZZZ, MR 36 #184.

Seiler, W.K., "Global Moduli for Elliptic Sufaces with a Section," Comp. Math. 62 (1987), 169-185.

Seiler, W.K., "Global Moduli for Polarized Surfaces," Comp. Math. 62 (1987) 187-213. Seller, W.K., "Deformation of WeierstraB Elliptic Surfaces/' Math. Annal. 1988

Sernesi, E., "Small deformations of global complete intersections, " B.U.M.I., IZ (1975), 138-146. ~ernesi, E., "Sullo schema di Hilbert," prepr{nt. ~erre, J.-P. , "Faisceaux algebrlques coherents, " Ann. Math., 61 (1955), 197-278. Serre, I.-P., "Tours dtarithmetique," Presses Universitaires de France, Paris (1970).

83

[sh]

[Sie i]

[Siu]

[Sp~v]

[s~]

[Tj~]

[To]

[Us]

[Wah]

[Way]

[We]

[Ya]

[Za]

[Zp]

[zp z]

~afarevitch, L , et al. , "Algebraic surfaces, " Proc. Steklov Inst. , A.M.S. Translation (1967), Providence. Siegel, C. L., "Meromorphe ~nktionen auf kompakten analytischen Mannigfaltigkeiten, Nachr. Akad. Wiss. G~ttingen, rnat/~-phys. Klasse 4 (1955), 71-77.

S[u, T. Y., "Every I<-3 surface is Ii~bler," Inv. Math., 73 (1983),

139-150o Spivak, Mo, "(A comprehensive introductfon to) Differential Geometry, " IIed., volo I-V, Publish or Perish, Berkeley (1979). Shwa, T. , "Stratification of local modul[ spaces of H[rzebruch mani- folds, " Rice University Studies 59 (1975), IZ9-146, MR 48 #436Z. Tjurina, G. IN., "Resolution of singularities of flat deformations of rational doublets," ~nktsional, Anal. i Prilozhen°, 4 (1) (1970), 77-83, MR 4Z #2031. Todorov, A° , "Applications of the K~hler-Einstein- Calabi-Yau metric to moduli of K3-surfaces," Invent. Math°, 81 (1980), Z51-Z66. Usui, So, "Deformations and local Torelli theorem for certain surfaces of general type," in Algebraic Geometry, Copenhagen ]978, Springer

L. NoM. 732 (1979), 605-623. V/ahl~ S° , "Deformations of plane curves with nodes and cusps, " Am. J.

Math., 96 (1974), 5Z9-577 Wavrik, J. J[o , "Obstructions to the existence of a space of moduli, " Global Analysis, Princeton Math. Series Z9 (1969), 403-414. ~Veil, A°, "Introduction a l'etude des varietes I~&'hleriennes, "

Hermann, Paris (1958). Yau, S. To, "Calabi's conjecture and some new results in algebraic geometry," Proc. Nat° Acad° Sci° U.S.A., 74 (1977), 1798-1799. Zarisk[, O., "Algebraic surfaces, 2nd ed. Erg. Math, 61,Springer,

Heidelberg (1971). Zappa, G. , "Sh[ Sisterni continu[ di curve sopra una rigata algebrica, "

Gior. Mat. Battaglini, 77 (1947). Zappa, G., "Sulla degenerazione delle superficie algebriche in sistemi di piani distinti, con applicaziont M lo studio de llerigate, " Atti Accad.

Naz. Lincei ~em. ,13 (1942)